KILLED
proof of input_3eR7Ic2pfA.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 37 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 307 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 180 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 60 ms]
            (26) typed CpxTrs
(27) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(28) CpxWeightedTrs
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CpxTypedWeightedTrs
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) NarrowingProof [BOTH BOUNDS(ID, ID), 77 ms]
        (34) CpxTypedWeightedCompleteTrs
        (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
        (36) CpxRNTS
        (37) InliningProof [UPPER BOUND(ID), 10.1 s]
        (38) CpxRNTS
        (39) SimplificationProof [BOTH BOUNDS(ID, ID), 43 ms]
        (40) CpxRNTS
        (41) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 9 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
        (48) CpxRNTS
        (49) ResultPropagationProof [UPPER BOUND(ID), 15 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
        (54) CpxRNTS
        (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
        (57) IntTrsBoundProof [UPPER BOUND(ID), 285 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (60) CpxRNTS
        (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (62) CpxRNTS
        (63) IntTrsBoundProof [UPPER BOUND(ID), 432 ms]
        (64) CpxRNTS
        (65) IntTrsBoundProof [UPPER BOUND(ID), 155 ms]
        (66) CpxRNTS
        (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
        (70) CpxRNTS
        (71) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (72) CpxRNTS
        (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
        (78) CpxRNTS
        (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (80) CpxRNTS
        (81) IntTrsBoundProof [UPPER BOUND(ID), 79 ms]
        (82) CpxRNTS
        (83) IntTrsBoundProof [UPPER BOUND(ID), 12 ms]
        (84) CpxRNTS
        (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (86) CpxRNTS
        (87) IntTrsBoundProof [UPPER BOUND(ID), 4952 ms]
        (88) CpxRNTS
        (89) IntTrsBoundProof [UPPER BOUND(ID), 1316 ms]
        (90) CpxRNTS
    (91) CompletionProof [UPPER BOUND(ID), 0 ms]
    (92) CpxTypedWeightedCompleteTrs
        (93) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (94) CpxRNTS
(95) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(96) CdtProblem
    (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 14 ms]
    (104) CdtProblem
    (105) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CpxRelTRS
        (107) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (108) CpxTRS
        (109) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CpxWeightedTrs
            (111) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (112) CpxTypedWeightedTrs
            (113) CompletionProof [UPPER BOUND(ID), 2 ms]
            (114) CpxTypedWeightedCompleteTrs
                (115) NarrowingProof [BOTH BOUNDS(ID, ID), 353 ms]
                (116) CpxTypedWeightedCompleteTrs
                (117) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms]
                (118) CpxRNTS
                (119) InliningProof [UPPER BOUND(ID), 4263 ms]
                (120) CpxRNTS
                (121) SimplificationProof [BOTH BOUNDS(ID, ID), 8 ms]
                (122) CpxRNTS
                (123) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (124) CpxRNTS
                (125) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (126) CpxRNTS
                (127) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
                (128) CpxRNTS
                (129) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
                (130) CpxRNTS
                (131) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (132) CpxRNTS
                (133) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
                (134) CpxRNTS
                (135) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
                (136) CpxRNTS
                (137) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (138) CpxRNTS
                (139) IntTrsBoundProof [UPPER BOUND(ID), 421 ms]
                (140) CpxRNTS
                (141) IntTrsBoundProof [UPPER BOUND(ID), 126 ms]
                (142) CpxRNTS
                (143) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (144) CpxRNTS
                (145) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
                (146) CpxRNTS
                (147) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
                (148) CpxRNTS
                (149) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (150) CpxRNTS
                (151) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
                (152) CpxRNTS
                (153) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
                (154) CpxRNTS
                (155) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (156) CpxRNTS
                (157) IntTrsBoundProof [UPPER BOUND(ID), 152 ms]
                (158) CpxRNTS
                (159) IntTrsBoundProof [UPPER BOUND(ID), 23 ms]
                (160) CpxRNTS
                (161) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (162) CpxRNTS
                (163) IntTrsBoundProof [UPPER BOUND(ID), 219 ms]
                (164) CpxRNTS
                (165) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
                (166) CpxRNTS
                (167) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (168) CpxRNTS
                (169) IntTrsBoundProof [UPPER BOUND(ID), 2979 ms]
                (170) CpxRNTS
                (171) IntTrsBoundProof [UPPER BOUND(ID), 2858 ms]
                (172) CpxRNTS
            (173) CompletionProof [UPPER BOUND(ID), 0 ms]
            (174) CpxTypedWeightedCompleteTrs
                (175) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
                (176) CpxRNTS
    (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
        (179) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CdtProblem
        (181) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CdtProblem
        (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (184) CdtProblem
        (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (186) CdtProblem
        (187) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (188) CdtProblem
        (189) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (190) CdtProblem
        (191) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (192) CdtProblem
        (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (194) CdtProblem
        (195) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (196) CdtProblem
        (197) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (198) CdtProblem
        (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (200) CdtProblem
        (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms]
        (202) CdtProblem
        (203) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (204) CdtProblem
        (205) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (206) CdtProblem
        (207) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (208) CdtProblem
        (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (210) CdtProblem
        (211) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (212) CdtProblem
        (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (214) CdtProblem
        (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (216) CdtProblem
        (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (218) CdtProblem
        (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (220) CdtProblem
        (221) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (222) CdtProblem
        (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (224) CdtProblem
        (225) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (226) CdtProblem
        (227) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (228) CdtProblem
        (229) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 143 ms]
        (230) CdtProblem
        (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (232) CdtProblem
        (233) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (234) CdtProblem
        (235) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (236) CdtProblem
        (237) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 102 ms]
        (238) CdtProblem
        (239) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (240) CdtProblem
        (241) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (242) CdtProblem
        (243) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (244) CdtProblem
        (245) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 113 ms]
        (246) CdtProblem
        (247) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (248) CdtProblem
        (249) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (250) CdtProblem
        (251) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (252) CdtProblem
        (253) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 139 ms]
        (254) CdtProblem
        (255) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
        (256) CdtProblem
        (257) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (258) CdtProblem
        (259) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (260) CdtProblem
        (261) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (262) CdtProblem
        (263) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (264) CdtProblem
        (265) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (266) CdtProblem
        (267) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (268) CdtProblem
        (269) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (270) CdtProblem
        (271) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (272) CdtProblem
        (273) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (274) CdtProblem
        (275) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (276) CdtProblem
        (277) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (278) CdtProblem
        (279) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (280) CdtProblem
        (281) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (282) CdtProblem
        (283) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (284) CdtProblem
        (285) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (286) CdtProblem
        (287) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (288) CdtProblem
        (289) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (290) CdtProblem
        (291) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (292) CdtProblem
        (293) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (294) CdtProblem
        (295) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (296) CdtProblem
        (297) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (298) CdtProblem
        (299) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (300) CdtProblem
        (301) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
        (302) CdtProblem
        (303) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (304) CdtProblem
        (305) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (306) CdtProblem
        (307) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (308) CdtProblem
        (309) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (310) CdtProblem
        (311) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (312) CdtProblem
        (313) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (314) CdtProblem
        (315) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms]
        (316) CdtProblem
        (317) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (318) CdtProblem
        (319) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (320) CdtProblem
        (321) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (322) CdtProblem
        (323) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (324) CdtProblem
        (325) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (326) CdtProblem
        (327) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (328) CdtProblem
        (329) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (330) CdtProblem
        (331) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (332) CdtProblem
        (333) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (334) CdtProblem
        (335) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (336) CdtProblem
        (337) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (338) CdtProblem
        (339) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (340) CdtProblem
        (341) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (342) CdtProblem
        (343) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (344) CdtProblem
        (345) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (346) CdtProblem
        (347) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (348) CdtProblem
        (349) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (350) CdtProblem
        (351) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (352) CdtProblem
        (353) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (354) CdtProblem
        (355) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (356) CdtProblem
        (357) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (358) CdtProblem
        (359) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (360) CdtProblem
        (361) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (362) CdtProblem
        (363) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (364) CdtProblem
        (365) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (366) CdtProblem
        (367) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (368) CdtProblem
        (369) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (370) CdtProblem
        (371) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (372) CdtProblem
        (373) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (374) CdtProblem
        (375) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (376) CdtProblem
        (377) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 258 ms]
        (378) CdtProblem
        (379) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (380) CdtProblem
        (381) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (382) CdtProblem
        (383) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (384) CdtProblem
        (385) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (386) CdtProblem
        (387) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (388) CdtProblem
        (389) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (390) CdtProblem
        (391) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (392) CdtProblem
        (393) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (394) CdtProblem
        (395) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (396) CdtProblem
        (397) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (398) CdtProblem
        (399) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 286 ms]
        (400) CdtProblem
        (401) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 270 ms]
        (402) CdtProblem
        (403) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (404) CdtProblem
        (405) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (406) CdtProblem
        (407) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (408) CdtProblem
        (409) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (410) CdtProblem
        (411) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (412) CdtProblem
        (413) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (414) CdtProblem
        (415) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (416) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0)))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0')))), plus(y, s(0')), plus(z, s(0')))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0)))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(true, z0, z1, z2) -> c5
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10
   LOW(z0, nil) -> c11
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, nil) -> c15
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(true, z0) -> c21
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(0, z0) -> c27
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   ISEMPTY(nil) -> c29
   ISEMPTY(cons(z0, z1)) -> c30
   HEAD(nil) -> c31
   HEAD(cons(z0, z1)) -> c32
   TAIL(nil) -> c33
   TAIL(cons(z0, z1)) -> c34
   GE(z0, 0) -> c35
   GE(0, s(z0)) -> c36
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   A -> c38
   A -> c39
S tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(true, z0, z1, z2) -> c5
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10
   LOW(z0, nil) -> c11
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, nil) -> c15
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(true, z0) -> c21
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(0, z0) -> c27
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   ISEMPTY(nil) -> c29
   ISEMPTY(cons(z0, z1)) -> c30
   HEAD(nil) -> c31
   HEAD(cons(z0, z1)) -> c32
   TAIL(nil) -> c33
   TAIL(cons(z0, z1)) -> c34
   GE(z0, 0) -> c35
   GE(0, s(z0)) -> c36
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   A -> c38
   A -> c39
K tuples:none
Defined Rule Symbols:   average_2, if_3, averIter_3, ifIter_4, append_2, low_2, if_low_3, high_2, if_high_3, quicksort_1, ifquick_2, plus_2, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   AVERAGE_2, IF_3, AVERITER_3, IFITER_4, APPEND_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, IFQUICK_2, PLUS_2, ISEMPTY_1, HEAD_1, TAIL_1, GE_2, A

Compound Symbols:   c1_2, c2_1, c3_1, c4_2, c5, c6_2, c7_2, c8_2, c9, c10, c11, c12_2, c13_1, c14_1, c15, c16_2, c17_1, c18_1, c19_1, c20_2, c21, c22_4, c23_4, c24_2, c25_4, c26_4, c27, c28_1, c29, c30, c31, c32, c33, c34, c35, c36, c37_1, c38, c39


----------------------------------------

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(true, z0, z1, z2) -> c5
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10
   LOW(z0, nil) -> c11
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, nil) -> c15
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(true, z0) -> c21
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(0, z0) -> c27
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   ISEMPTY(nil) -> c29
   ISEMPTY(cons(z0, z1)) -> c30
   HEAD(nil) -> c31
   HEAD(cons(z0, z1)) -> c32
   TAIL(nil) -> c33
   TAIL(cons(z0, z1)) -> c34
   GE(z0, 0) -> c35
   GE(0, s(z0)) -> c36
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   A -> c38
   A -> c39

The (relative) TRS S consists of the following rules:

   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c

Rewrite Strategy: INNERMOST
----------------------------------------

(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(true, z0, z1, z2) -> c5
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10
   LOW(z0, nil) -> c11
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, nil) -> c15
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(true, z0) -> c21
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(0', z0) -> c27
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   ISEMPTY(nil) -> c29
   ISEMPTY(cons(z0, z1)) -> c30
   HEAD(nil) -> c31
   HEAD(cons(z0, z1)) -> c32
   TAIL(nil) -> c33
   TAIL(cons(z0, z1)) -> c34
   GE(z0, 0') -> c35
   GE(0', s(z0)) -> c36
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   A -> c38
   A -> c39

The (relative) TRS S consists of the following rules:

   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0', z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c

Rewrite Strategy: INNERMOST
----------------------------------------

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28

----------------------------------------

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ge, GE, AVERITER, plus, PLUS, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
ge < AVERITER
ge < LOW
ge < HIGH
ge < low
ge < high
ge < averIter
GE < AVERITER
GE < LOW
GE < HIGH
plus < AVERITER
PLUS < AVERITER
plus < averIter
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28


Generator Equations:
gen_0':s:nil:cons:app:error21_40(0) <=> 0'
gen_0':s:nil:cons:app:error21_40(+(x, 1)) <=> s(gen_0':s:nil:cons:app:error21_40(x))
gen_c35:c36:c3722_40(0) <=> c35
gen_c35:c36:c3722_40(+(x, 1)) <=> c37(gen_c35:c36:c3722_40(x))
gen_c27:c2823_40(0) <=> c27
gen_c27:c2823_40(+(x, 1)) <=> c28(gen_c27:c2823_40(x))


The following defined symbols remain to be analysed:
ge, GE, AVERITER, plus, PLUS, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
ge < AVERITER
ge < LOW
ge < HIGH
ge < low
ge < high
ge < averIter
GE < AVERITER
GE < LOW
GE < HIGH
plus < AVERITER
PLUS < AVERITER
plus < averIter
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) -> true, rt in Omega(0)

Induction Base:
ge(gen_0':s:nil:cons:app:error21_40(0), gen_0':s:nil:cons:app:error21_40(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_0':s:nil:cons:app:error21_40(+(n25_40, 1)), gen_0':s:nil:cons:app:error21_40(+(n25_40, 1))) ->_R^Omega(0)
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28


Lemmas:
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s:nil:cons:app:error21_40(0) <=> 0'
gen_0':s:nil:cons:app:error21_40(+(x, 1)) <=> s(gen_0':s:nil:cons:app:error21_40(x))
gen_c35:c36:c3722_40(0) <=> c35
gen_c35:c36:c3722_40(+(x, 1)) <=> c37(gen_c35:c36:c3722_40(x))
gen_c27:c2823_40(0) <=> c27
gen_c27:c2823_40(+(x, 1)) <=> c28(gen_c27:c2823_40(x))


The following defined symbols remain to be analysed:
GE, AVERITER, plus, PLUS, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
GE < AVERITER
GE < LOW
GE < HIGH
plus < AVERITER
PLUS < AVERITER
plus < averIter
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_0':s:nil:cons:app:error21_40(n795_40), gen_0':s:nil:cons:app:error21_40(n795_40)) -> gen_c35:c36:c3722_40(n795_40), rt in Omega(1 + n795_40)

Induction Base:
GE(gen_0':s:nil:cons:app:error21_40(0), gen_0':s:nil:cons:app:error21_40(0)) ->_R^Omega(1)
c35

Induction Step:
GE(gen_0':s:nil:cons:app:error21_40(+(n795_40, 1)), gen_0':s:nil:cons:app:error21_40(+(n795_40, 1))) ->_R^Omega(1)
c37(GE(gen_0':s:nil:cons:app:error21_40(n795_40), gen_0':s:nil:cons:app:error21_40(n795_40))) ->_IH
c37(gen_c35:c36:c3722_40(c796_40))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Complex Obligation (BEST)

----------------------------------------

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28


Lemmas:
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s:nil:cons:app:error21_40(0) <=> 0'
gen_0':s:nil:cons:app:error21_40(+(x, 1)) <=> s(gen_0':s:nil:cons:app:error21_40(x))
gen_c35:c36:c3722_40(0) <=> c35
gen_c35:c36:c3722_40(+(x, 1)) <=> c37(gen_c35:c36:c3722_40(x))
gen_c27:c2823_40(0) <=> c27
gen_c27:c2823_40(+(x, 1)) <=> c28(gen_c27:c2823_40(x))


The following defined symbols remain to be analysed:
GE, AVERITER, plus, PLUS, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
GE < AVERITER
GE < LOW
GE < HIGH
plus < AVERITER
PLUS < AVERITER
plus < averIter
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(21)
BOUNDS(n^1, INF)

----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28


Lemmas:
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) -> true, rt in Omega(0)
GE(gen_0':s:nil:cons:app:error21_40(n795_40), gen_0':s:nil:cons:app:error21_40(n795_40)) -> gen_c35:c36:c3722_40(n795_40), rt in Omega(1 + n795_40)


Generator Equations:
gen_0':s:nil:cons:app:error21_40(0) <=> 0'
gen_0':s:nil:cons:app:error21_40(+(x, 1)) <=> s(gen_0':s:nil:cons:app:error21_40(x))
gen_c35:c36:c3722_40(0) <=> c35
gen_c35:c36:c3722_40(+(x, 1)) <=> c37(gen_c35:c36:c3722_40(x))
gen_c27:c2823_40(0) <=> c27
gen_c27:c2823_40(+(x, 1)) <=> c28(gen_c27:c2823_40(x))


The following defined symbols remain to be analysed:
plus, AVERITER, PLUS, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
plus < AVERITER
PLUS < AVERITER
plus < averIter
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s:nil:cons:app:error21_40(n1878_40), gen_0':s:nil:cons:app:error21_40(b)) -> gen_0':s:nil:cons:app:error21_40(+(n1878_40, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s:nil:cons:app:error21_40(0), gen_0':s:nil:cons:app:error21_40(b)) ->_R^Omega(0)
gen_0':s:nil:cons:app:error21_40(b)

Induction Step:
plus(gen_0':s:nil:cons:app:error21_40(+(n1878_40, 1)), gen_0':s:nil:cons:app:error21_40(b)) ->_R^Omega(0)
s(plus(gen_0':s:nil:cons:app:error21_40(n1878_40), gen_0':s:nil:cons:app:error21_40(b))) ->_IH
s(gen_0':s:nil:cons:app:error21_40(+(b, c1879_40)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28


Lemmas:
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) -> true, rt in Omega(0)
GE(gen_0':s:nil:cons:app:error21_40(n795_40), gen_0':s:nil:cons:app:error21_40(n795_40)) -> gen_c35:c36:c3722_40(n795_40), rt in Omega(1 + n795_40)
plus(gen_0':s:nil:cons:app:error21_40(n1878_40), gen_0':s:nil:cons:app:error21_40(b)) -> gen_0':s:nil:cons:app:error21_40(+(n1878_40, b)), rt in Omega(0)


Generator Equations:
gen_0':s:nil:cons:app:error21_40(0) <=> 0'
gen_0':s:nil:cons:app:error21_40(+(x, 1)) <=> s(gen_0':s:nil:cons:app:error21_40(x))
gen_c35:c36:c3722_40(0) <=> c35
gen_c35:c36:c3722_40(+(x, 1)) <=> c37(gen_c35:c36:c3722_40(x))
gen_c27:c2823_40(0) <=> c27
gen_c27:c2823_40(+(x, 1)) <=> c28(gen_c27:c2823_40(x))


The following defined symbols remain to be analysed:
PLUS, AVERITER, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
PLUS < AVERITER
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s:nil:cons:app:error21_40(n3923_40), gen_0':s:nil:cons:app:error21_40(b)) -> gen_c27:c2823_40(n3923_40), rt in Omega(1 + n3923_40)

Induction Base:
PLUS(gen_0':s:nil:cons:app:error21_40(0), gen_0':s:nil:cons:app:error21_40(b)) ->_R^Omega(1)
c27

Induction Step:
PLUS(gen_0':s:nil:cons:app:error21_40(+(n3923_40, 1)), gen_0':s:nil:cons:app:error21_40(b)) ->_R^Omega(1)
c28(PLUS(gen_0':s:nil:cons:app:error21_40(n3923_40), gen_0':s:nil:cons:app:error21_40(b))) ->_IH
c28(gen_c27:c2823_40(c3924_40))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Obligation:
Innermost TRS:
Rules:
AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
IFITER(true, z0, z1, z2) -> c5
IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z0, s(s(s(0')))))
IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z1, s(0')))
IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0'))), PLUS(z2, s(0')))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10
LOW(z0, nil) -> c11
LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
HIGH(z0, nil) -> c15
HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
IFQUICK(true, z0) -> c21
IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
PLUS(0', z0) -> c27
PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
ISEMPTY(nil) -> c29
ISEMPTY(cons(z0, z1)) -> c30
HEAD(nil) -> c31
HEAD(cons(z0, z1)) -> c32
TAIL(nil) -> c33
TAIL(cons(z0, z1)) -> c34
GE(z0, 0') -> c35
GE(0', s(z0)) -> c36
GE(s(z0), s(z1)) -> c37(GE(z0, z1))
A -> c38
A -> c39
average(z0, z1) -> if(ge(z0, z1), z0, z1)
if(true, z0, z1) -> averIter(z1, z0, z1)
if(false, z0, z1) -> averIter(z0, z1, z0)
averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
ifIter(true, z0, z1, z2) -> z2
ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0')))), plus(z1, s(0')), plus(z2, s(0')))
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
low(z0, nil) -> nil
low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
high(z0, nil) -> nil
high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
quicksort(z0) -> ifquick(isempty(z0), z0)
ifquick(true, z0) -> nil
ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
isempty(nil) -> true
isempty(cons(z0, z1)) -> false
head(nil) -> error
head(cons(z0, z1)) -> z0
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
a -> b
a -> c

Types:
AVERAGE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c1
c1 :: c2:c3 -> c35:c36:c37 -> c1
IF :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c2:c3
ge :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> true:false
GE :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c35:c36:c37
true :: true:false
c2 :: c4 -> c2:c3
AVERITER :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c4
false :: true:false
c3 :: c4 -> c2:c3
c4 :: c5:c6:c7:c8 -> c35:c36:c37 -> c4
IFITER :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c4 -> c27:c28 -> c5:c6:c7:c8
plus :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
s :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
0' :: 0':s:nil:cons:app:error
PLUS :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c27:c28
c7 :: c4 -> c27:c28 -> c5:c6:c7:c8
c8 :: c4 -> c27:c28 -> c5:c6:c7:c8
APPEND :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c9:c10
nil :: 0':s:nil:cons:app:error
c9 :: c9:c10
cons :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
c10 :: c9:c10
LOW :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c11:c12
c11 :: c11:c12
c12 :: c13:c14 -> c35:c36:c37 -> c11:c12
IF_LOW :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c13:c14
c13 :: c11:c12 -> c13:c14
c14 :: c11:c12 -> c13:c14
HIGH :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c15:c16
c15 :: c15:c16
c16 :: c17:c18:c19 -> c35:c36:c37 -> c15:c16
IF_HIGH :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> c17:c18:c19
c17 :: c15:c16 -> c17:c18:c19
c18 :: c1 -> c17:c18:c19
c19 :: c15:c16 -> c17:c18:c19
QUICKSORT :: 0':s:nil:cons:app:error -> c20
c20 :: c21:c22:c23:c24:c25:c26 -> c29:c30 -> c20
IFQUICK :: true:false -> 0':s:nil:cons:app:error -> c21:c22:c23:c24:c25:c26
isempty :: 0':s:nil:cons:app:error -> true:false
ISEMPTY :: 0':s:nil:cons:app:error -> c29:c30
c21 :: c21:c22:c23:c24:c25:c26
c22 :: c9:c10 -> c20 -> c11:c12 -> c31:c32 -> c21:c22:c23:c24:c25:c26
quicksort :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
low :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
head :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
tail :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
high :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
HEAD :: 0':s:nil:cons:app:error -> c31:c32
c23 :: c9:c10 -> c20 -> c11:c12 -> c33:c34 -> c21:c22:c23:c24:c25:c26
TAIL :: 0':s:nil:cons:app:error -> c33:c34
c24 :: c9:c10 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c25 :: c9:c10 -> c20 -> c15:c16 -> c31:c32 -> c21:c22:c23:c24:c25:c26
c26 :: c9:c10 -> c20 -> c15:c16 -> c33:c34 -> c21:c22:c23:c24:c25:c26
c27 :: c27:c28
c28 :: c27:c28 -> c27:c28
c29 :: c29:c30
c30 :: c29:c30
c31 :: c31:c32
c32 :: c31:c32
c33 :: c33:c34
c34 :: c33:c34
c35 :: c35:c36:c37
c36 :: c35:c36:c37
c37 :: c35:c36:c37 -> c35:c36:c37
A :: c38:c39
c38 :: c38:c39
c39 :: c38:c39
average :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
averIter :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifIter :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
append :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
app :: 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_low :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
if_high :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
ifquick :: true:false -> 0':s:nil:cons:app:error -> 0':s:nil:cons:app:error
error :: 0':s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c
hole_c11_40 :: c1
hole_0':s:nil:cons:app:error2_40 :: 0':s:nil:cons:app:error
hole_c2:c33_40 :: c2:c3
hole_c35:c36:c374_40 :: c35:c36:c37
hole_true:false5_40 :: true:false
hole_c46_40 :: c4
hole_c5:c6:c7:c87_40 :: c5:c6:c7:c8
hole_c27:c288_40 :: c27:c28
hole_c9:c109_40 :: c9:c10
hole_c11:c1210_40 :: c11:c12
hole_c13:c1411_40 :: c13:c14
hole_c15:c1612_40 :: c15:c16
hole_c17:c18:c1913_40 :: c17:c18:c19
hole_c2014_40 :: c20
hole_c21:c22:c23:c24:c25:c2615_40 :: c21:c22:c23:c24:c25:c26
hole_c29:c3016_40 :: c29:c30
hole_c31:c3217_40 :: c31:c32
hole_c33:c3418_40 :: c33:c34
hole_c38:c3919_40 :: c38:c39
hole_b:c20_40 :: b:c
gen_0':s:nil:cons:app:error21_40 :: Nat -> 0':s:nil:cons:app:error
gen_c35:c36:c3722_40 :: Nat -> c35:c36:c37
gen_c27:c2823_40 :: Nat -> c27:c28


Lemmas:
ge(gen_0':s:nil:cons:app:error21_40(n25_40), gen_0':s:nil:cons:app:error21_40(n25_40)) -> true, rt in Omega(0)
GE(gen_0':s:nil:cons:app:error21_40(n795_40), gen_0':s:nil:cons:app:error21_40(n795_40)) -> gen_c35:c36:c3722_40(n795_40), rt in Omega(1 + n795_40)
plus(gen_0':s:nil:cons:app:error21_40(n1878_40), gen_0':s:nil:cons:app:error21_40(b)) -> gen_0':s:nil:cons:app:error21_40(+(n1878_40, b)), rt in Omega(0)
PLUS(gen_0':s:nil:cons:app:error21_40(n3923_40), gen_0':s:nil:cons:app:error21_40(b)) -> gen_c27:c2823_40(n3923_40), rt in Omega(1 + n3923_40)


Generator Equations:
gen_0':s:nil:cons:app:error21_40(0) <=> 0'
gen_0':s:nil:cons:app:error21_40(+(x, 1)) <=> s(gen_0':s:nil:cons:app:error21_40(x))
gen_c35:c36:c3722_40(0) <=> c35
gen_c35:c36:c3722_40(+(x, 1)) <=> c37(gen_c35:c36:c3722_40(x))
gen_c27:c2823_40(0) <=> c27
gen_c27:c2823_40(+(x, 1)) <=> c28(gen_c27:c2823_40(x))


The following defined symbols remain to be analysed:
AVERITER, LOW, HIGH, QUICKSORT, quicksort, low, high, averIter

They will be analysed ascendingly in the following order:
LOW < QUICKSORT
HIGH < QUICKSORT
quicksort < QUICKSORT
low < QUICKSORT
high < QUICKSORT
low < quicksort
high < quicksort

----------------------------------------

(27) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y) [1]
   if(true, x, y) -> averIter(y, x, y) [1]
   if(false, x, y) -> averIter(x, y, x) [1]
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z) [1]
   ifIter(true, x, y, z) -> z [1]
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0))) [1]
   append(nil, y) -> y [1]
   append(cons(n, x), y) -> cons(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x)) [1]
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x)) [1]
   if_low(true, n, cons(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x)) [1]
   if_high(false, n, cons(m, x)) -> high(n, x) [1]
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x)) [1]
   quicksort(x) -> ifquick(isempty(x), x) [1]
   ifquick(true, x) -> nil [1]
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x))))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   isempty(nil) -> true [1]
   isempty(cons(n, x)) -> false [1]
   head(nil) -> error [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(29) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y) [1]
   if(true, x, y) -> averIter(y, x, y) [1]
   if(false, x, y) -> averIter(x, y, x) [1]
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z) [1]
   ifIter(true, x, y, z) -> z [1]
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0))) [1]
   append(nil, y) -> y [1]
   append(cons(n, x), y) -> cons(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x)) [1]
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x)) [1]
   if_low(true, n, cons(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x)) [1]
   if_high(false, n, cons(m, x)) -> high(n, x) [1]
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x)) [1]
   quicksort(x) -> ifquick(isempty(x), x) [1]
   ifquick(true, x) -> nil [1]
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x))))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   isempty(nil) -> true [1]
   isempty(cons(n, x)) -> false [1]
   head(nil) -> error [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]

The TRS has the following type information:
average :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
if :: true:false -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
ge :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> true:false
true :: true:false
averIter :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
false :: true:false
ifIter :: true:false -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
plus :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
s :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
0 :: 0:s:nil:cons:app:error
append :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
nil :: 0:s:nil:cons:app:error
cons :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
app :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
low :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
if_low :: true:false -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
high :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
if_high :: true:false -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
quicksort :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
ifquick :: true:false -> 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
isempty :: 0:s:nil:cons:app:error -> true:false
head :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
tail :: 0:s:nil:cons:app:error -> 0:s:nil:cons:app:error
error :: 0:s:nil:cons:app:error
a :: b:c
b :: b:c
c :: b:c

Rewrite Strategy: INNERMOST
----------------------------------------

(31) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   a

(c) The following functions are completely defined:

   ge_2
   plus_2
   quicksort_1
   low_2
   head_1
   tail_1
   high_2
   isempty_1
   if_high_3
   average_2
   if_low_3
   ifquick_2
   if_3
   averIter_3
   append_2
   ifIter_4

Due to the following rules being added:

   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   low(v0, v1) -> null_low [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   high(v0, v1) -> null_high [0]
   isempty(v0) -> null_isempty [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   append(v0, v1) -> null_append [0]
   ifquick(v0, v1) -> null_ifquick [0]
   if(v0, v1, v2) -> null_if [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]

And the following fresh constants:    null_ge, null_plus, null_low, null_head, null_tail, null_high, null_isempty, null_if_high, null_if_low, null_append, null_ifquick, null_if, null_ifIter

----------------------------------------

(32)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y) [1]
   if(true, x, y) -> averIter(y, x, y) [1]
   if(false, x, y) -> averIter(x, y, x) [1]
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z) [1]
   ifIter(true, x, y, z) -> z [1]
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0))) [1]
   append(nil, y) -> y [1]
   append(cons(n, x), y) -> cons(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x)) [1]
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x)) [1]
   if_low(true, n, cons(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x)) [1]
   if_high(false, n, cons(m, x)) -> high(n, x) [1]
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x)) [1]
   quicksort(x) -> ifquick(isempty(x), x) [1]
   ifquick(true, x) -> nil [1]
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x))))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   isempty(nil) -> true [1]
   isempty(cons(n, x)) -> false [1]
   head(nil) -> error [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   low(v0, v1) -> null_low [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   high(v0, v1) -> null_high [0]
   isempty(v0) -> null_isempty [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   append(v0, v1) -> null_append [0]
   ifquick(v0, v1) -> null_ifquick [0]
   if(v0, v1, v2) -> null_if [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]

The TRS has the following type information:
average :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
if :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
ge :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> true:false:null_ge:null_isempty
true :: true:false:null_ge:null_isempty
averIter :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
false :: true:false:null_ge:null_isempty
ifIter :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
plus :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
s :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
0 :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
append :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
nil :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
cons :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
app :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
low :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
if_low :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
high :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
if_high :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
quicksort :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
ifquick :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
isempty :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> true:false:null_ge:null_isempty
head :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
tail :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
error :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
a :: b:c
b :: b:c
c :: b:c
null_ge :: true:false:null_ge:null_isempty
null_plus :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_low :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_head :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_tail :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_high :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_isempty :: true:false:null_ge:null_isempty
null_if_high :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_if_low :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_append :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_ifquick :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_if :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_ifIter :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter

Rewrite Strategy: INNERMOST
----------------------------------------

(33) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(34)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   average(x, 0) -> if(true, x, 0) [2]
   average(0, s(y')) -> if(false, 0, s(y')) [2]
   average(s(x'), s(y'')) -> if(ge(x', y''), s(x'), s(y'')) [2]
   average(x, y) -> if(null_ge, x, y) [1]
   if(true, x, y) -> averIter(y, x, y) [1]
   if(false, x, y) -> averIter(x, y, x) [1]
   averIter(x, 0, z) -> ifIter(true, x, 0, z) [2]
   averIter(0, s(y1), z) -> ifIter(false, 0, s(y1), z) [2]
   averIter(s(x''), s(y2), z) -> ifIter(ge(x'', y2), s(x''), s(y2), z) [2]
   averIter(x, y, z) -> ifIter(null_ge, x, y, z) [1]
   ifIter(true, x, y, z) -> z [1]
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0))) [1]
   append(nil, y) -> y [1]
   append(cons(n, x), y) -> cons(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(0, cons(m, x)) -> if_low(true, 0, cons(m, x)) [2]
   low(s(y3), cons(0, x)) -> if_low(false, s(y3), cons(0, x)) [2]
   low(s(y4), cons(s(x14), x)) -> if_low(ge(x14, y4), s(y4), cons(s(x14), x)) [2]
   low(n, cons(m, x)) -> if_low(null_ge, n, cons(m, x)) [1]
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x)) [1]
   if_low(true, n, cons(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(0, cons(m, x)) -> if_high(true, 0, cons(m, x)) [2]
   high(s(y5), cons(0, x)) -> if_high(false, s(y5), cons(0, x)) [2]
   high(s(y6), cons(s(x15), x)) -> if_high(ge(x15, y6), s(y6), cons(s(x15), x)) [2]
   high(n, cons(m, x)) -> if_high(null_ge, n, cons(m, x)) [1]
   if_high(false, n, cons(m, x)) -> high(n, x) [1]
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x)) [1]
   quicksort(nil) -> ifquick(true, nil) [2]
   quicksort(cons(n', x16)) -> ifquick(false, cons(n', x16)) [2]
   quicksort(x) -> ifquick(null_isempty, x) [1]
   ifquick(true, x) -> nil [1]
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x))))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   isempty(nil) -> true [1]
   isempty(cons(n, x)) -> false [1]
   head(nil) -> error [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   low(v0, v1) -> null_low [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   high(v0, v1) -> null_high [0]
   isempty(v0) -> null_isempty [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   append(v0, v1) -> null_append [0]
   ifquick(v0, v1) -> null_ifquick [0]
   if(v0, v1, v2) -> null_if [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]

The TRS has the following type information:
average :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
if :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
ge :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> true:false:null_ge:null_isempty
true :: true:false:null_ge:null_isempty
averIter :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
false :: true:false:null_ge:null_isempty
ifIter :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
plus :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
s :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
0 :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
append :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
nil :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
cons :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
app :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
low :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
if_low :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
high :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
if_high :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
quicksort :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
ifquick :: true:false:null_ge:null_isempty -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
isempty :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> true:false:null_ge:null_isempty
head :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
tail :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter -> 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
error :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
a :: b:c
b :: b:c
c :: b:c
null_ge :: true:false:null_ge:null_isempty
null_plus :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_low :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_head :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_tail :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_high :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_isempty :: true:false:null_ge:null_isempty
null_if_high :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_if_low :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_append :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_ifquick :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_if :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter
null_ifIter :: 0:s:nil:cons:app:error:null_plus:null_low:null_head:null_tail:null_high:null_if_high:null_if_low:null_append:null_ifquick:null_if:null_ifIter

Rewrite Strategy: INNERMOST
----------------------------------------

(35) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   false => 1
   0 => 0
   nil => 2
   error => 1
   b => 0
   c => 1
   null_ge => 0
   null_plus => 0
   null_low => 0
   null_head => 0
   null_tail => 0
   null_high => 0
   null_isempty => 0
   null_if_high => 0
   null_if_low => 0
   null_append => 0
   null_ifquick => 0
   null_if => 0
   null_ifIter => 0

----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> y :|: z' = 2, z'' = y, y >= 0
   append(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(x'', y2), 1 + x'', 1 + y2, z) :|: z1 = z, z >= 0, z' = 1 + x'', z'' = 1 + y2, x'' >= 0, y2 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, x, 0, z) :|: z'' = 0, z1 = z, z >= 0, z' = x, x >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + y1, z) :|: y1 >= 0, z1 = z, z >= 0, z'' = 1 + y1, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   average(z', z'') -{ 2 }-> if(ge(x', y''), 1 + x', 1 + y'') :|: z' = 1 + x', x' >= 0, y'' >= 0, z'' = 1 + y''
   average(z', z'') -{ 2 }-> if(2, x, 0) :|: z'' = 0, z' = x, x >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + y') :|: y' >= 0, z' = 0, z'' = 1 + y'
   average(z', z'') -{ 1 }-> if(0, x, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   high(z', z'') -{ 2 }-> if_high(ge(x15, y6), 1 + y6, 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, y6 >= 0, z' = 1 + y6, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + y5, 1 + 0 + x) :|: y5 >= 0, x >= 0, z' = 1 + y5, z'' = 1 + 0 + x
   high(z', z'') -{ 1 }-> if_high(0, n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: n >= 0, z' = n, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   if(z', z'', z1) -{ 1 }-> averIter(x, y, x) :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(y, x, y) :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   ifIter(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(x, 1 + (1 + (1 + 0))), plus(y, 1 + 0), plus(z, 1 + 0)) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   if_high(z', z'', z1) -{ 1 }-> high(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(head(x), tail(x))), 1 + tail(x) + quicksort(high(head(x), tail(x)))) :|: x >= 0, z'' = x, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, x >= 0, z'' = x
   ifquick(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   low(z', z'') -{ 2 }-> if_low(ge(x14, y4), 1 + y4, 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' = 1 + y4, y4 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + y3, 1 + 0 + x) :|: x >= 0, y3 >= 0, z' = 1 + y3, z'' = 1 + 0 + x
   low(z', z'') -{ 1 }-> if_low(0, n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: n >= 0, z' = n, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, x) :|: z' = x, x >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0


----------------------------------------

(37) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   append(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   append(z', z'') -{ 1 }-> y :|: z' = 2, z'' = y, y >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   head(z') -{ 1 }-> 1 :|: z' = 2
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0

----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> y :|: z' = 2, z'' = y, y >= 0
   append(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(x'', y2), 1 + x'', 1 + y2, z) :|: z1 = z, z >= 0, z' = 1 + x'', z'' = 1 + y2, x'' >= 0, y2 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, x, 0, z) :|: z'' = 0, z1 = z, z >= 0, z' = x, x >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + y1, z) :|: y1 >= 0, z1 = z, z >= 0, z'' = 1 + y1, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   average(z', z'') -{ 2 }-> if(ge(x', y''), 1 + x', 1 + y'') :|: z' = 1 + x', x' >= 0, y'' >= 0, z'' = 1 + y''
   average(z', z'') -{ 2 }-> if(2, x, 0) :|: z'' = 0, z' = x, x >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + y') :|: y' >= 0, z' = 0, z'' = 1 + y'
   average(z', z'') -{ 1 }-> if(0, x, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   high(z', z'') -{ 2 }-> if_high(ge(x15, y6), 1 + y6, 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, y6 >= 0, z' = 1 + y6, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + y5, 1 + 0 + x) :|: y5 >= 0, x >= 0, z' = 1 + y5, z'' = 1 + 0 + x
   high(z', z'') -{ 1 }-> if_high(0, n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: n >= 0, z' = n, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   if(z', z'', z1) -{ 1 }-> averIter(x, y, x) :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(y, x, y) :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   ifIter(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(x, 1 + (1 + (1 + 0))), plus(y, 1 + 0), plus(z, 1 + 0)) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   if_high(z', z'', z1) -{ 1 }-> high(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0', n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0', x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0, n2 >= 0, x = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0, x = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, x = 2, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, x = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0 >= 0, x = v0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0 >= 0, x = v0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, v0 >= 0, x = v0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', x = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, x = 2, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, v0'' >= 0, x = v0'', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, v0'' >= 0, x = v0'', x = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, v0'' >= 0, x = v0'', v01 >= 0, x = v01
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, n1 >= 0, x = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0', n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0', x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, n >= 0, x = 1 + n + x', x' >= 0, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, v0'' >= 0, x = v0'', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', x = 2, v0'' >= 0, x = v0'', v01 >= 0, x = v01
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x = 1 + n + x', x' >= 0, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x = 1 + n + x', x' >= 0, x = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x = 1 + n + x', x' >= 0, v01 >= 0, x = v01
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0, n'' >= 0, x = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0, x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', n' >= 0, x = 1 + n' + x'', x'' >= 0, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', x = 2, n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', x = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', x = 2, v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0'
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', x = 2, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', x = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', x = 2, v01 >= 0, x = v01
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n' >= 0, x = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', x = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, n >= 0, x = 1 + n + x', x' >= 0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0'', n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0''
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, x = 2, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, n >= 0, x = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, x = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: x >= 0, z'' = x, z' = 1, v0 >= 0, x = v0, v0' >= 0, x = v0', v0'' >= 0, x = v0'', v01 >= 0, x = v01, v02 >= 0, x = v02
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, x >= 0, z'' = x
   ifquick(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   low(z', z'') -{ 2 }-> if_low(ge(x14, y4), 1 + y4, 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' = 1 + y4, y4 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + y3, 1 + 0 + x) :|: x >= 0, y3 >= 0, z' = 1 + y3, z'' = 1 + 0 + x
   low(z', z'') -{ 1 }-> if_low(0, n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: n >= 0, z' = n, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, x) :|: z' = x, x >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0


----------------------------------------

(39) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0


----------------------------------------

(41) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { tail }
   { isempty }
   { append }
   { ge }
   { head }
   { plus }
   { a }
   { averIter, ifIter }
   { low, if_low }
   { if }
   { average }
   { high, if_high }
   { ifquick, quicksort }

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {tail}, {isempty}, {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}

----------------------------------------

(43) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {tail}, {isempty}, {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}

----------------------------------------

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {tail}, {isempty}, {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: ?, size: O(n^1) [z']

----------------------------------------

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {isempty}, {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(49) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {isempty}, {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {isempty}, {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: ?, size: O(1) [2]

----------------------------------------

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]

----------------------------------------

(55) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]

----------------------------------------

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: append
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {append}, {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: ?, size: O(n^1) [1 + z' + z'']

----------------------------------------

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: append
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']

----------------------------------------

(61) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']

----------------------------------------

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {ge}, {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: ?, size: O(1) [2]

----------------------------------------

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 2 }-> if(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 2 }-> if_high(ge(x15, z' - 1), 1 + (z' - 1), 1 + (1 + x15) + x) :|: x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 2 }-> if_low(ge(x14, z' - 1), 1 + (z' - 1), 1 + (1 + x14) + x) :|: z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]

----------------------------------------

(67) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]

----------------------------------------

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {head}, {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: ?, size: O(n^1) [z']

----------------------------------------

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(73) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z' + z''

----------------------------------------

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {plus}, {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: ?, size: O(n^1) [z' + z'']

----------------------------------------

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

----------------------------------------

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(z'', 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

----------------------------------------

(79) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 4 + z'' + z1 + z2 }-> averIter(s3, s4, s5) :|: s3 >= 0, s3 <= z'' + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z1 + (1 + 0), s5 >= 0, s5 <= z2 + (1 + 0), z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

----------------------------------------

(81) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: a
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 4 + z'' + z1 + z2 }-> averIter(s3, s4, s5) :|: s3 >= 0, s3 <= z'' + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z1 + (1 + 0), s5 >= 0, s5 <= z2 + (1 + 0), z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {a}, {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: ?, size: O(1) [1]

----------------------------------------

(83) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: a
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 4 + z'' + z1 + z2 }-> averIter(s3, s4, s5) :|: s3 >= 0, s3 <= z'' + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z1 + (1 + 0), s5 >= 0, s5 <= z2 + (1 + 0), z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(85) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 4 + z'' + z1 + z2 }-> averIter(s3, s4, s5) :|: s3 >= 0, s3 <= z'' + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z1 + (1 + 0), s5 >= 0, s5 <= z2 + (1 + 0), z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(87) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: averIter
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: ifIter
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 4 + z'' + z1 + z2 }-> averIter(s3, s4, s5) :|: s3 >= 0, s3 <= z'' + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z1 + (1 + 0), s5 >= 0, s5 <= z2 + (1 + 0), z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]
  averIter: runtime: ?, size: INF
  ifIter: runtime: ?, size: INF

----------------------------------------

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: averIter
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> z'' :|: z' = 2, z'' >= 0
   append(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   averIter(z', z'', z1) -{ 3 + z' }-> ifIter(s', 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s' >= 0, s' <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(2, z', 0, z1) :|: z'' = 0, z1 >= 0, z' >= 0
   averIter(z', z'', z1) -{ 2 }-> ifIter(1, 0, 1 + (z'' - 1), z1) :|: z'' - 1 >= 0, z1 >= 0, z' = 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   average(z', z'') -{ 3 + z' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   average(z', z'') -{ 2 }-> if(2, z', 0) :|: z'' = 0, z' >= 0
   average(z', z'') -{ 2 }-> if(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   average(z', z'') -{ 1 }-> if(0, z', z'') :|: z' >= 0, z'' >= 0
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: z' >= 0
   high(z', z'') -{ 4 + x15 }-> if_high(s1, 1 + (z' - 1), 1 + (1 + x15) + x) :|: s1 >= 0, s1 <= 2, x >= 0, z'' = 1 + (1 + x15) + x, z' - 1 >= 0, x15 >= 0
   high(z', z'') -{ 2 }-> if_high(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   high(z', z'') -{ 2 }-> if_high(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   high(z', z'') -{ 1 }-> if_high(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if(z', z'', z1) -{ 1 }-> averIter(z'', z1, z'') :|: z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(z1, z'', z1) :|: z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifIter(z', z'', z1, z2) -{ 4 + z'' + z1 + z2 }-> averIter(s3, s4, s5) :|: s3 >= 0, s3 <= z'' + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z1 + (1 + 0), s5 >= 0, s5 <= z2 + (1 + 0), z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_high(z', z'', z1) -{ 1 }-> high(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(z'', x) :|: z'' >= 0, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(z'', x) :|: z'' >= 0, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + x1 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, x'')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', x3))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, n2 >= 0, z'' = 1 + n2 + x3, x3 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + x2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, x1)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + x1 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(n', 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(n, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 6 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(1, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, x1))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + x'' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, x')), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, x2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, n1 >= 0, z'' = 1 + n1 + x2, x2 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + x1 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, x'')), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 5 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 2)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, x''))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + x' + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, x1))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, n'' >= 0, z'' = 1 + n'' + x1, x1 >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + x'' + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 4 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 2 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, x''))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, n' >= 0, z'' = 1 + n' + x'', x'' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 2))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(n, 0))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, x'))) :|: z'' >= 0, z' = 1, z'' = 2, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 3 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(1, 0))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, x'))) :|: z'' >= 0, z' = 1, n >= 0, z'' = 1 + n + x', x' >= 0
   ifquick(z', z'') -{ 2 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 2))) :|: z'' >= 0, z' = 1, z'' = 2
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(0, 0)), 1 + 0 + quicksort(high(0, 0))) :|: z'' >= 0, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, z'' >= 0
   ifquick(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: z' >= 0
   low(z', z'') -{ 4 + x14 }-> if_low(s'', 1 + (z' - 1), 1 + (1 + x14) + x) :|: s'' >= 0, s'' <= 2, z'' = 1 + (1 + x14) + x, x >= 0, x14 >= 0, z' - 1 >= 0
   low(z', z'') -{ 2 }-> if_low(2, 0, 1 + m + x) :|: z'' = 1 + m + x, x >= 0, z' = 0, m >= 0
   low(z', z'') -{ 2 }-> if_low(1, 1 + (z' - 1), 1 + 0 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0
   low(z', z'') -{ 1 }-> if_low(0, z', 1 + m + x) :|: z' >= 0, z'' = 1 + m + x, x >= 0, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: z' >= 0, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quicksort(z') -{ 2 }-> ifquick(2, 2) :|: z' = 2
   quicksort(z') -{ 2 }-> ifquick(1, 1 + n' + x16) :|: n' >= 0, z' = 1 + n' + x16, x16 >= 0
   quicksort(z') -{ 1 }-> ifquick(0, z') :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {averIter,ifIter}, {low,if_low}, {if}, {average}, {high,if_high}, {ifquick,quicksort}
Previous analysis results are:
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  append: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  a: runtime: O(1) [1], size: O(1) [1]
  averIter: runtime: INF, size: INF
  ifIter: runtime: ?, size: INF

----------------------------------------

(91) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   append(v0, v1) -> null_append [0]
   low(v0, v1) -> null_low [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   high(v0, v1) -> null_high [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2) -> null_if [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]
   ifquick(v0, v1) -> null_ifquick [0]

And the following fresh constants:    null_append, null_low, null_if_low, null_high, null_if_high, null_plus, null_isempty, null_head, null_tail, null_ge, null_if, null_ifIter, null_ifquick

----------------------------------------

(92)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y) [1]
   if(true, x, y) -> averIter(y, x, y) [1]
   if(false, x, y) -> averIter(x, y, x) [1]
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z) [1]
   ifIter(true, x, y, z) -> z [1]
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0))) [1]
   append(nil, y) -> y [1]
   append(cons(n, x), y) -> cons(n, app(x, y)) [1]
   low(n, nil) -> nil [1]
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x)) [1]
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x)) [1]
   if_low(true, n, cons(m, x)) -> low(n, x) [1]
   high(n, nil) -> nil [1]
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x)) [1]
   if_high(false, n, cons(m, x)) -> high(n, x) [1]
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x)) [1]
   quicksort(x) -> ifquick(isempty(x), x) [1]
   ifquick(true, x) -> nil [1]
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x))))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   isempty(nil) -> true [1]
   isempty(cons(n, x)) -> false [1]
   head(nil) -> error [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> b [1]
   a -> c [1]
   append(v0, v1) -> null_append [0]
   low(v0, v1) -> null_low [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   high(v0, v1) -> null_high [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2) -> null_if [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]
   ifquick(v0, v1) -> null_ifquick [0]

The TRS has the following type information:
average :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
if :: true:false:null_isempty:null_ge -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
ge :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> true:false:null_isempty:null_ge
true :: true:false:null_isempty:null_ge
averIter :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
false :: true:false:null_isempty:null_ge
ifIter :: true:false:null_isempty:null_ge -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
plus :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
s :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
0 :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
append :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
nil :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
cons :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
app :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
low :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
if_low :: true:false:null_isempty:null_ge -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
high :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
if_high :: true:false:null_isempty:null_ge -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
quicksort :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
ifquick :: true:false:null_isempty:null_ge -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
isempty :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> true:false:null_isempty:null_ge
head :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
tail :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick -> 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
error :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
a :: b:c
b :: b:c
c :: b:c
null_append :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_low :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_if_low :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_high :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_if_high :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_plus :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_isempty :: true:false:null_isempty:null_ge
null_head :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_tail :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_ge :: true:false:null_isempty:null_ge
null_if :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_ifIter :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick
null_ifquick :: 0:s:nil:cons:app:error:null_append:null_low:null_if_low:null_high:null_if_high:null_plus:null_head:null_tail:null_if:null_ifIter:null_ifquick

Rewrite Strategy: INNERMOST
----------------------------------------

(93) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   false => 1
   0 => 0
   nil => 2
   error => 1
   b => 0
   c => 1
   null_append => 0
   null_low => 0
   null_if_low => 0
   null_high => 0
   null_if_high => 0
   null_plus => 0
   null_isempty => 0
   null_head => 0
   null_tail => 0
   null_ge => 0
   null_if => 0
   null_ifIter => 0
   null_ifquick => 0

----------------------------------------

(94)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   append(z', z'') -{ 1 }-> y :|: z' = 2, z'' = y, y >= 0
   append(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   append(z', z'') -{ 1 }-> 1 + n + (1 + x + y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   averIter(z', z'', z1) -{ 1 }-> ifIter(ge(x, y), x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   average(z', z'') -{ 1 }-> if(ge(x, y), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 2
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   high(z', z'') -{ 1 }-> if_high(ge(m, n), n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   high(z', z'') -{ 1 }-> 2 :|: n >= 0, z' = n, z'' = 2
   high(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   if(z', z'', z1) -{ 1 }-> averIter(x, y, x) :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1) -{ 1 }-> averIter(y, x, y) :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   ifIter(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   ifIter(z', z'', z1, z2) -{ 1 }-> averIter(plus(x, 1 + (1 + (1 + 0))), plus(y, 1 + 0), plus(z, 1 + 0)) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   ifIter(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   if_high(z', z'', z1) -{ 1 }-> high(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   if_high(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if_high(z', z'', z1) -{ 1 }-> 1 + average(m, m) + high(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 1 }-> low(n, x) :|: n >= 0, z'' = n, z' = 2, x >= 0, z1 = 1 + m + x, m >= 0
   if_low(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if_low(z', z'', z1) -{ 1 }-> 1 + m + low(n, x) :|: n >= 0, z'' = n, x >= 0, z1 = 1 + m + x, z' = 1, m >= 0
   ifquick(z', z'') -{ 1 }-> append(quicksort(low(head(x), tail(x))), 1 + tail(x) + quicksort(high(head(x), tail(x)))) :|: x >= 0, z'' = x, z' = 1
   ifquick(z', z'') -{ 1 }-> 2 :|: z' = 2, x >= 0, z'' = x
   ifquick(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   isempty(z') -{ 1 }-> 2 :|: z' = 2
   isempty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   isempty(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   low(z', z'') -{ 1 }-> if_low(ge(m, n), n, 1 + m + x) :|: n >= 0, z'' = 1 + m + x, x >= 0, z' = n, m >= 0
   low(z', z'') -{ 1 }-> 2 :|: n >= 0, z' = n, z'' = 2
   low(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quicksort(z') -{ 1 }-> ifquick(isempty(x), x) :|: z' = x, x >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 2 :|: z' = 2
   tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(95) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(true, z0, z1, z2) -> c5
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10
   LOW(z0, nil) -> c11
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, nil) -> c15
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(true, z0) -> c21
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(0, z0) -> c27
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   ISEMPTY(nil) -> c29
   ISEMPTY(cons(z0, z1)) -> c30
   HEAD(nil) -> c31
   HEAD(cons(z0, z1)) -> c32
   TAIL(nil) -> c33
   TAIL(cons(z0, z1)) -> c34
   GE(z0, 0) -> c35
   GE(0, s(z0)) -> c36
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   A -> c38
   A -> c39
S tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(true, z0, z1, z2) -> c5
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10
   LOW(z0, nil) -> c11
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, nil) -> c15
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(true, z0) -> c21
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(0, z0) -> c27
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   ISEMPTY(nil) -> c29
   ISEMPTY(cons(z0, z1)) -> c30
   HEAD(nil) -> c31
   HEAD(cons(z0, z1)) -> c32
   TAIL(nil) -> c33
   TAIL(cons(z0, z1)) -> c34
   GE(z0, 0) -> c35
   GE(0, s(z0)) -> c36
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   A -> c38
   A -> c39
K tuples:none
Defined Rule Symbols:   average_2, if_3, averIter_3, ifIter_4, append_2, low_2, if_low_3, high_2, if_high_3, quicksort_1, ifquick_2, plus_2, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   AVERAGE_2, IF_3, AVERITER_3, IFITER_4, APPEND_2, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, IFQUICK_2, PLUS_2, ISEMPTY_1, HEAD_1, TAIL_1, GE_2, A

Compound Symbols:   c1_2, c2_1, c3_1, c4_2, c5, c6_2, c7_2, c8_2, c9, c10, c11, c12_2, c13_1, c14_1, c15, c16_2, c17_1, c18_1, c19_1, c20_2, c21, c22_4, c23_4, c24_2, c25_4, c26_4, c27, c28_1, c29, c30, c31, c32, c33, c34, c35, c36, c37_1, c38, c39


----------------------------------------

(97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 18 trailing nodes:
   GE(z0, 0) -> c35
   HIGH(z0, nil) -> c15
   LOW(z0, nil) -> c11
   HEAD(cons(z0, z1)) -> c32
   HEAD(nil) -> c31
   IFQUICK(true, z0) -> c21
   TAIL(nil) -> c33
   PLUS(0, z0) -> c27
   IFQUICK(false, z0) -> c24(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), TAIL(z0))
   TAIL(cons(z0, z1)) -> c34
   APPEND(cons(z0, z1), z2) -> c10
   APPEND(nil, z0) -> c9
   A -> c38
   GE(0, s(z0)) -> c36
   A -> c39
   ISEMPTY(nil) -> c29
   IFITER(true, z0, z1, z2) -> c5
   ISEMPTY(cons(z0, z1)) -> c30

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
S tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0), ISEMPTY(z0))
   IFQUICK(false, z0) -> c22(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c23(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)), TAIL(z0))
   IFQUICK(false, z0) -> c25(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), HEAD(z0))
   IFQUICK(false, z0) -> c26(APPEND(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0))))), QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)), TAIL(z0))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   average_2, if_3, averIter_3, ifIter_4, append_2, low_2, if_low_3, high_2, if_high_3, quicksort_1, ifquick_2, plus_2, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   AVERAGE_2, IF_3, AVERITER_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, QUICKSORT_1, IFQUICK_2, PLUS_2, GE_2

Compound Symbols:   c1_2, c2_1, c3_1, c4_2, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c20_2, c22_4, c23_4, c25_4, c26_4, c28_1, c37_1


----------------------------------------

(99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing tuple parts
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
S tuples:
   AVERAGE(z0, z1) -> c1(IF(ge(z0, z1), z0, z1), GE(z0, z1))
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
K tuples:none
Defined Rule Symbols:   average_2, if_3, averIter_3, ifIter_4, append_2, low_2, if_low_3, high_2, if_high_3, quicksort_1, ifquick_2, plus_2, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   AVERAGE_2, IF_3, AVERITER_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c1_2, c2_1, c3_1, c4_2, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2


----------------------------------------

(101) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> b
   a -> c
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   average_2, if_3, averIter_3, ifIter_4, append_2, low_2, if_low_3, high_2, if_high_3, quicksort_1, ifquick_2, plus_2, isempty_1, head_1, tail_1, ge_2, a

Defined Pair Symbols:   IF_3, AVERITER_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2

Compound Symbols:   c2_1, c3_1, c4_2, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1


----------------------------------------

(103) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   quicksort(z0) -> ifquick(isempty(z0), z0)
   ifquick(true, z0) -> nil
   ifquick(false, z0) -> append(quicksort(low(head(z0), tail(z0))), cons(tail(z0), quicksort(high(head(z0), tail(z0)))))
   a -> b
   a -> c

----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, AVERITER_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2

Compound Symbols:   c2_1, c3_1, c4_2, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1


----------------------------------------

(105) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(106)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))

The (relative) TRS S consists of the following rules:

   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))

Rewrite Strategy: INNERMOST
----------------------------------------

(107) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(108)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(109) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(110)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1)) [1]
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0)) [1]
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1)) [1]
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0))))) [1]
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0))) [1]
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0))) [1]
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2)) [1]
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2)) [1]
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2)) [1]
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c37(GE(z0, z1)) [1]
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0)) [1]
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1)) [1]
   AVERAGE(z0, z1) -> c5(GE(z0, z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> s(plus(z0, z1)) [0]
   isempty(nil) -> true [0]
   isempty(cons(z0, z1)) -> false [0]
   low(z0, nil) -> nil [0]
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2)) [0]
   head(nil) -> error [0]
   head(cons(z0, z1)) -> z0 [0]
   tail(nil) -> nil [0]
   tail(cons(z0, z1)) -> z1 [0]
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) [0]
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2) [0]
   high(z0, nil) -> nil [0]
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2)) [0]
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2) [0]
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2)) [0]
   average(z0, z1) -> if(ge(z0, z1), z0, z1) [0]
   if(true, z0, z1) -> averIter(z1, z0, z1) [0]
   if(false, z0, z1) -> averIter(z0, z1, z0) [0]
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2) [0]
   ifIter(true, z0, z1, z2) -> z2 [0]
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(111) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(112)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1)) [1]
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0)) [1]
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1)) [1]
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0))))) [1]
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0))) [1]
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0))) [1]
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2)) [1]
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2)) [1]
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2)) [1]
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c37(GE(z0, z1)) [1]
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0)) [1]
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1)) [1]
   AVERAGE(z0, z1) -> c5(GE(z0, z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> s(plus(z0, z1)) [0]
   isempty(nil) -> true [0]
   isempty(cons(z0, z1)) -> false [0]
   low(z0, nil) -> nil [0]
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2)) [0]
   head(nil) -> error [0]
   head(cons(z0, z1)) -> z0 [0]
   tail(nil) -> nil [0]
   tail(cons(z0, z1)) -> z1 [0]
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) [0]
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2) [0]
   high(z0, nil) -> nil [0]
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2)) [0]
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2) [0]
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2)) [0]
   average(z0, z1) -> if(ge(z0, z1), z0, z1) [0]
   if(true, z0, z1) -> averIter(z1, z0, z1) [0]
   if(false, z0, z1) -> averIter(z0, z1, z0) [0]
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2) [0]
   ifIter(true, z0, z1, z2) -> z2 [0]
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))) [0]

The TRS has the following type information:
IF :: true:false -> 0:s:error -> 0:s:error -> c2:c3:c37
true :: true:false
c2 :: c4 -> c2:c3:c37
AVERITER :: 0:s:error -> 0:s:error -> 0:s:error -> c4
false :: true:false
c3 :: c4 -> c2:c3:c37
c4 :: c6:c7:c8 -> c2:c3:c37 -> c4
IFITER :: true:false -> 0:s:error -> 0:s:error -> 0:s:error -> c6:c7:c8
ge :: 0:s:error -> 0:s:error -> true:false
GE :: 0:s:error -> 0:s:error -> c2:c3:c37
c6 :: c4 -> c28 -> c6:c7:c8
plus :: 0:s:error -> 0:s:error -> 0:s:error
s :: 0:s:error -> 0:s:error
0 :: 0:s:error
PLUS :: 0:s:error -> 0:s:error -> c28
c7 :: c4 -> c28 -> c6:c7:c8
c8 :: c4 -> c28 -> c6:c7:c8
LOW :: 0:s:error -> cons:nil -> c12
cons :: 0:s:error -> cons:nil -> cons:nil
c12 :: c13:c14 -> c2:c3:c37 -> c12
IF_LOW :: true:false -> 0:s:error -> cons:nil -> c13:c14
c13 :: c12 -> c13:c14
c14 :: c12 -> c13:c14
HIGH :: 0:s:error -> cons:nil -> c16
c16 :: c17:c18:c19 -> c2:c3:c37 -> c16
IF_HIGH :: true:false -> 0:s:error -> cons:nil -> c17:c18:c19
c17 :: c16 -> c17:c18:c19
c18 :: c5 -> c17:c18:c19
AVERAGE :: 0:s:error -> 0:s:error -> c5
c19 :: c16 -> c17:c18:c19
c28 :: c28 -> c28
c37 :: c2:c3:c37 -> c2:c3:c37
QUICKSORT :: cons:nil -> c20
c20 :: c22:c23:c25:c26 -> c20
IFQUICK :: true:false -> cons:nil -> c22:c23:c25:c26
isempty :: cons:nil -> true:false
c22 :: c20 -> c12 -> c22:c23:c25:c26
low :: 0:s:error -> cons:nil -> cons:nil
head :: cons:nil -> 0:s:error
tail :: cons:nil -> cons:nil
c23 :: c20 -> c12 -> c22:c23:c25:c26
c25 :: c20 -> c16 -> c22:c23:c25:c26
high :: 0:s:error -> cons:nil -> cons:nil
c26 :: c20 -> c16 -> c22:c23:c25:c26
c5 :: c2:c3:c37 -> c5
nil :: cons:nil
if_low :: true:false -> 0:s:error -> cons:nil -> cons:nil
error :: 0:s:error
if_high :: true:false -> 0:s:error -> cons:nil -> cons:nil
average :: 0:s:error -> 0:s:error -> 0:s:error
if :: true:false -> 0:s:error -> 0:s:error -> 0:s:error
averIter :: 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error
ifIter :: true:false -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error

Rewrite Strategy: INNERMOST
----------------------------------------

(113) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   IF_3
   AVERITER_3
   IFITER_4
   LOW_2
   IF_LOW_3
   HIGH_2
   IF_HIGH_3
   PLUS_2
   GE_2
   QUICKSORT_1
   IFQUICK_2
   AVERAGE_2

(c) The following functions are completely defined:

   ge_2
   plus_2
   isempty_1
   low_2
   head_1
   tail_1
   if_low_3
   high_2
   if_high_3
   average_2
   if_3
   averIter_3
   ifIter_4

Due to the following rules being added:

   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   low(v0, v1) -> nil [0]
   head(v0) -> null_head [0]
   tail(v0) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]
   high(v0, v1) -> nil [0]
   if_high(v0, v1, v2) -> nil [0]
   average(v0, v1) -> null_average [0]
   if(v0, v1, v2) -> null_if [0]
   averIter(v0, v1, v2) -> null_averIter [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]

And the following fresh constants:    null_ge, null_plus, null_isempty, null_head, null_average, null_if, null_averIter, null_ifIter, const, const1, const2, const3, const4, const5, const6, const7, const8, const9, const10

----------------------------------------

(114)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1)) [1]
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0)) [1]
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1)) [1]
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0))))) [1]
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0))) [1]
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0))) [1]
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2)) [1]
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2)) [1]
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2)) [1]
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c37(GE(z0, z1)) [1]
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0)) [1]
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1)) [1]
   AVERAGE(z0, z1) -> c5(GE(z0, z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> s(plus(z0, z1)) [0]
   isempty(nil) -> true [0]
   isempty(cons(z0, z1)) -> false [0]
   low(z0, nil) -> nil [0]
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2)) [0]
   head(nil) -> error [0]
   head(cons(z0, z1)) -> z0 [0]
   tail(nil) -> nil [0]
   tail(cons(z0, z1)) -> z1 [0]
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) [0]
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2) [0]
   high(z0, nil) -> nil [0]
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2)) [0]
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2) [0]
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2)) [0]
   average(z0, z1) -> if(ge(z0, z1), z0, z1) [0]
   if(true, z0, z1) -> averIter(z1, z0, z1) [0]
   if(false, z0, z1) -> averIter(z0, z1, z0) [0]
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2) [0]
   ifIter(true, z0, z1, z2) -> z2 [0]
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))) [0]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   low(v0, v1) -> nil [0]
   head(v0) -> null_head [0]
   tail(v0) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]
   high(v0, v1) -> nil [0]
   if_high(v0, v1, v2) -> nil [0]
   average(v0, v1) -> null_average [0]
   if(v0, v1, v2) -> null_if [0]
   averIter(v0, v1, v2) -> null_averIter [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]

The TRS has the following type information:
IF :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c2:c3:c37
true :: true:false:null_ge:null_isempty
c2 :: c4 -> c2:c3:c37
AVERITER :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c4
false :: true:false:null_ge:null_isempty
c3 :: c4 -> c2:c3:c37
c4 :: c6:c7:c8 -> c2:c3:c37 -> c4
IFITER :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c6:c7:c8
ge :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> true:false:null_ge:null_isempty
GE :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c2:c3:c37
c6 :: c4 -> c28 -> c6:c7:c8
plus :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
s :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
0 :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
PLUS :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c28
c7 :: c4 -> c28 -> c6:c7:c8
c8 :: c4 -> c28 -> c6:c7:c8
LOW :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c12
cons :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
c12 :: c13:c14 -> c2:c3:c37 -> c12
IF_LOW :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c13:c14
c13 :: c12 -> c13:c14
c14 :: c12 -> c13:c14
HIGH :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c16
c16 :: c17:c18:c19 -> c2:c3:c37 -> c16
IF_HIGH :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c17:c18:c19
c17 :: c16 -> c17:c18:c19
c18 :: c5 -> c17:c18:c19
AVERAGE :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c5
c19 :: c16 -> c17:c18:c19
c28 :: c28 -> c28
c37 :: c2:c3:c37 -> c2:c3:c37
QUICKSORT :: cons:nil -> c20
c20 :: c22:c23:c25:c26 -> c20
IFQUICK :: true:false:null_ge:null_isempty -> cons:nil -> c22:c23:c25:c26
isempty :: cons:nil -> true:false:null_ge:null_isempty
c22 :: c20 -> c12 -> c22:c23:c25:c26
low :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
head :: cons:nil -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
tail :: cons:nil -> cons:nil
c23 :: c20 -> c12 -> c22:c23:c25:c26
c25 :: c20 -> c16 -> c22:c23:c25:c26
high :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
c26 :: c20 -> c16 -> c22:c23:c25:c26
c5 :: c2:c3:c37 -> c5
nil :: cons:nil
if_low :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
error :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
if_high :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
average :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
if :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
averIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
ifIter :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_ge :: true:false:null_ge:null_isempty
null_plus :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_isempty :: true:false:null_ge:null_isempty
null_head :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_average :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_if :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_averIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_ifIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
const :: c2:c3:c37
const1 :: c4
const2 :: c6:c7:c8
const3 :: c28
const4 :: c12
const5 :: c13:c14
const6 :: c16
const7 :: c17:c18:c19
const8 :: c5
const9 :: c20
const10 :: c22:c23:c25:c26

Rewrite Strategy: INNERMOST
----------------------------------------

(115) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(116)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1)) [1]
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0)) [1]
   AVERITER(z0, 0, z2) -> c4(IFITER(true, z0, 0, z2), GE(z0, 0)) [1]
   AVERITER(0, s(z0'), z2) -> c4(IFITER(false, 0, s(z0'), z2), GE(0, s(z0'))) [1]
   AVERITER(s(z0''), s(z1'), z2) -> c4(IFITER(ge(z0'', z1'), s(z0''), s(z1'), z2), GE(s(z0''), s(z1'))) [1]
   AVERITER(z0, z1, z2) -> c4(IFITER(null_ge, z0, z1, z2), GE(z0, z1)) [1]
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0))))) [1]
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0))) [1]
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0))) [1]
   LOW(0, cons(z1, z2)) -> c12(IF_LOW(true, 0, cons(z1, z2)), GE(z1, 0)) [1]
   LOW(s(z040), cons(0, z2)) -> c12(IF_LOW(false, s(z040), cons(0, z2)), GE(0, s(z040))) [1]
   LOW(s(z1''), cons(s(z041), z2)) -> c12(IF_LOW(ge(z041, z1''), s(z1''), cons(s(z041), z2)), GE(s(z041), s(z1''))) [1]
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(null_ge, z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2)) [1]
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2)) [1]
   HIGH(0, cons(z1, z2)) -> c16(IF_HIGH(true, 0, cons(z1, z2)), GE(z1, 0)) [1]
   HIGH(s(z042), cons(0, z2)) -> c16(IF_HIGH(false, s(z042), cons(0, z2)), GE(0, s(z042))) [1]
   HIGH(s(z11), cons(s(z043), z2)) -> c16(IF_HIGH(ge(z043, z11), s(z11), cons(s(z043), z2)), GE(s(z043), s(z11))) [1]
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(null_ge, z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2)) [1]
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c37(GE(z0, z1)) [1]
   QUICKSORT(nil) -> c20(IFQUICK(true, nil)) [1]
   QUICKSORT(cons(z044, z12)) -> c20(IFQUICK(false, cons(z044, z12))) [1]
   QUICKSORT(z0) -> c20(IFQUICK(null_isempty, z0)) [1]
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0)) [1]
   AVERAGE(0, s(z061)) -> c5(IF(false, 0, s(z061))) [1]
   AVERAGE(s(z062), s(z119)) -> c5(IF(ge(z062, z119), s(z062), s(z119))) [1]
   AVERAGE(z0, z1) -> c5(IF(null_ge, z0, z1)) [1]
   AVERAGE(z0, z1) -> c5(GE(z0, z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> s(plus(z0, z1)) [0]
   isempty(nil) -> true [0]
   isempty(cons(z0, z1)) -> false [0]
   low(z0, nil) -> nil [0]
   low(0, cons(z1, z2)) -> if_low(true, 0, cons(z1, z2)) [0]
   low(s(z063), cons(0, z2)) -> if_low(false, s(z063), cons(0, z2)) [0]
   low(s(z120), cons(s(z064), z2)) -> if_low(ge(z064, z120), s(z120), cons(s(z064), z2)) [0]
   low(z0, cons(z1, z2)) -> if_low(null_ge, z0, cons(z1, z2)) [0]
   head(nil) -> error [0]
   head(cons(z0, z1)) -> z0 [0]
   tail(nil) -> nil [0]
   tail(cons(z0, z1)) -> z1 [0]
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) [0]
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2) [0]
   high(z0, nil) -> nil [0]
   high(0, cons(z1, z2)) -> if_high(true, 0, cons(z1, z2)) [0]
   high(s(z065), cons(0, z2)) -> if_high(false, s(z065), cons(0, z2)) [0]
   high(s(z121), cons(s(z066), z2)) -> if_high(ge(z066, z121), s(z121), cons(s(z066), z2)) [0]
   high(z0, cons(z1, z2)) -> if_high(null_ge, z0, cons(z1, z2)) [0]
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2) [0]
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2)) [0]
   average(z0, 0) -> if(true, z0, 0) [0]
   average(0, s(z067)) -> if(false, 0, s(z067)) [0]
   average(s(z068), s(z122)) -> if(ge(z068, z122), s(z068), s(z122)) [0]
   average(z0, z1) -> if(null_ge, z0, z1) [0]
   if(true, z0, z1) -> averIter(z1, z0, z1) [0]
   if(false, z0, z1) -> averIter(z0, z1, z0) [0]
   averIter(z0, 0, z2) -> ifIter(true, z0, 0, z2) [0]
   averIter(0, s(z069), z2) -> ifIter(false, 0, s(z069), z2) [0]
   averIter(s(z070), s(z123), z2) -> ifIter(ge(z070, z123), s(z070), s(z123), z2) [0]
   averIter(z0, z1, z2) -> ifIter(null_ge, z0, z1, z2) [0]
   ifIter(true, z0, z1, z2) -> z2 [0]
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))) [0]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   low(v0, v1) -> nil [0]
   head(v0) -> null_head [0]
   tail(v0) -> nil [0]
   if_low(v0, v1, v2) -> nil [0]
   high(v0, v1) -> nil [0]
   if_high(v0, v1, v2) -> nil [0]
   average(v0, v1) -> null_average [0]
   if(v0, v1, v2) -> null_if [0]
   averIter(v0, v1, v2) -> null_averIter [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]

The TRS has the following type information:
IF :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c2:c3:c37
true :: true:false:null_ge:null_isempty
c2 :: c4 -> c2:c3:c37
AVERITER :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c4
false :: true:false:null_ge:null_isempty
c3 :: c4 -> c2:c3:c37
c4 :: c6:c7:c8 -> c2:c3:c37 -> c4
IFITER :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c6:c7:c8
ge :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> true:false:null_ge:null_isempty
GE :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c2:c3:c37
c6 :: c4 -> c28 -> c6:c7:c8
plus :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
s :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
0 :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
PLUS :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c28
c7 :: c4 -> c28 -> c6:c7:c8
c8 :: c4 -> c28 -> c6:c7:c8
LOW :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c12
cons :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
c12 :: c13:c14 -> c2:c3:c37 -> c12
IF_LOW :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c13:c14
c13 :: c12 -> c13:c14
c14 :: c12 -> c13:c14
HIGH :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c16
c16 :: c17:c18:c19 -> c2:c3:c37 -> c16
IF_HIGH :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> c17:c18:c19
c17 :: c16 -> c17:c18:c19
c18 :: c5 -> c17:c18:c19
AVERAGE :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c5
c19 :: c16 -> c17:c18:c19
c28 :: c28 -> c28
c37 :: c2:c3:c37 -> c2:c3:c37
QUICKSORT :: cons:nil -> c20
c20 :: c22:c23:c25:c26 -> c20
IFQUICK :: true:false:null_ge:null_isempty -> cons:nil -> c22:c23:c25:c26
isempty :: cons:nil -> true:false:null_ge:null_isempty
c22 :: c20 -> c12 -> c22:c23:c25:c26
low :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
head :: cons:nil -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
tail :: cons:nil -> cons:nil
c23 :: c20 -> c12 -> c22:c23:c25:c26
c25 :: c20 -> c16 -> c22:c23:c25:c26
high :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
c26 :: c20 -> c16 -> c22:c23:c25:c26
c5 :: c2:c3:c37 -> c5
nil :: cons:nil
if_low :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
error :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
if_high :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil -> cons:nil
average :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
if :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
averIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
ifIter :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_ge :: true:false:null_ge:null_isempty
null_plus :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_isempty :: true:false:null_ge:null_isempty
null_head :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_average :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_if :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_averIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_ifIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
const :: c2:c3:c37
const1 :: c4
const2 :: c6:c7:c8
const3 :: c28
const4 :: c12
const5 :: c13:c14
const6 :: c16
const7 :: c17:c18:c19
const8 :: c5
const9 :: c20
const10 :: c22:c23:c25:c26

Rewrite Strategy: INNERMOST
----------------------------------------

(117) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   false => 1
   0 => 0
   nil => 0
   error => 1
   null_ge => 0
   null_plus => 0
   null_isempty => 0
   null_head => 0
   null_average => 0
   null_if => 0
   null_averIter => 0
   null_ifIter => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0
   const5 => 0
   const6 => 0
   const7 => 0
   const8 => 0
   const9 => 0
   const10 => 0

----------------------------------------

(118)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z062, z119), 1 + z062, 1 + z119) :|: z = 1 + z062, z062 >= 0, z119 >= 0, z' = 1 + z119
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z0, 0) :|: z = z0, z0 >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + z061) :|: z' = 1 + z061, z061 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z0'', z1'), 1 + z0'', 1 + z1', z2) + GE(1 + z0'', 1 + z1') :|: z'' = z2, z' = 1 + z1', z1' >= 0, z0'' >= 0, z2 >= 0, z = 1 + z0''
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z0, 0, z2) + GE(z0, 0) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + z0', z2) + GE(0, 1 + z0') :|: z'' = z2, z0' >= 0, z = 0, z' = 1 + z0', z2 >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z0, z1, z2) + GE(z0, z1) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z11), 1 + z11, 1 + (1 + z043) + z2) + GE(1 + z043, 1 + z11) :|: z' = 1 + (1 + z043) + z2, z11 >= 0, z043 >= 0, z2 >= 0, z = 1 + z11
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + z042, 1 + 0 + z2) + GE(0, 1 + z042) :|: z042 >= 0, z = 1 + z042, z2 >= 0, z' = 1 + 0 + z2
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z0, 1 + z1 + z2) + GE(z1, z0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z0, z1, z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z1, z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z0, 1 + (1 + (1 + 0))) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z1, 1 + 0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z2, 1 + 0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(head(z0), tail(z0))) + LOW(head(z0), tail(z0)) :|: z = 1, z0 >= 0, z' = z0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(head(z0), tail(z0))) + HIGH(head(z0), tail(z0)) :|: z = 1, z0 >= 0, z' = z0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z1''), 1 + z1'', 1 + (1 + z041) + z2) + GE(1 + z041, 1 + z1'') :|: z' = 1 + (1 + z041) + z2, z = 1 + z1'', z041 >= 0, z2 >= 0, z1'' >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + z040, 1 + 0 + z2) + GE(0, 1 + z040) :|: z040 >= 0, z2 >= 0, z = 1 + z040, z' = 1 + 0 + z2
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z0, 1 + z1 + z2) + GE(z1, z0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z0) :|: z = z0, z0 >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z070, z123), 1 + z070, 1 + z123, z2) :|: z'' = z2, z123 >= 0, z' = 1 + z123, z070 >= 0, z2 >= 0, z = 1 + z070
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z0, 0, z2) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + z069, z2) :|: z'' = z2, z' = 1 + z069, z069 >= 0, z = 0, z2 >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z0, z1, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   average(z, z') -{ 0 }-> if(ge(z068, z122), 1 + z068, 1 + z122) :|: z = 1 + z068, z122 >= 0, z' = 1 + z122, z068 >= 0
   average(z, z') -{ 0 }-> if(2, z0, 0) :|: z = z0, z0 >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + z067) :|: z' = 1 + z067, z = 0, z067 >= 0
   average(z, z') -{ 0 }-> if(0, z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   average(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   high(z, z') -{ 0 }-> if_high(ge(z066, z121), 1 + z121, 1 + (1 + z066) + z2) :|: z121 >= 0, z = 1 + z121, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + z065, 1 + 0 + z2) :|: z = 1 + z065, z065 >= 0, z2 >= 0, z' = 1 + 0 + z2
   high(z, z') -{ 0 }-> if_high(0, z0, 1 + z1 + z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 0 }-> averIter(z0, z1, z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> averIter(z1, z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z2 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   low(z, z') -{ 0 }-> if_low(ge(z064, z120), 1 + z120, 1 + (1 + z064) + z2) :|: z120 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z = 1 + z120, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + z063, 1 + 0 + z2) :|: z063 >= 0, z = 1 + z063, z2 >= 0, z' = 1 + 0 + z2
   low(z, z') -{ 0 }-> if_low(0, z0, 1 + z1 + z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + plus(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


----------------------------------------

(119) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0

----------------------------------------

(120)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z062, z119), 1 + z062, 1 + z119) :|: z = 1 + z062, z062 >= 0, z119 >= 0, z' = 1 + z119
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z0, 0) :|: z = z0, z0 >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + z061) :|: z' = 1 + z061, z061 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z0'', z1'), 1 + z0'', 1 + z1', z2) + GE(1 + z0'', 1 + z1') :|: z'' = z2, z' = 1 + z1', z1' >= 0, z0'' >= 0, z2 >= 0, z = 1 + z0''
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z0, 0, z2) + GE(z0, 0) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + z0', z2) + GE(0, 1 + z0') :|: z'' = z2, z0' >= 0, z = 0, z' = 1 + z0', z2 >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z0, z1, z2) + GE(z0, z1) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z11), 1 + z11, 1 + (1 + z043) + z2) + GE(1 + z043, 1 + z11) :|: z' = 1 + (1 + z043) + z2, z11 >= 0, z043 >= 0, z2 >= 0, z = 1 + z11
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + z042, 1 + 0 + z2) + GE(0, 1 + z042) :|: z042 >= 0, z = 1 + z042, z2 >= 0, z' = 1 + 0 + z2
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z0, 1 + z1 + z2) + GE(z1, z0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z0, z1, z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z1, z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z0, 1 + (1 + (1 + 0))) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z1, 1 + 0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z2, 1 + 0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z0 = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1'', v0 >= 0, z0 = v0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0 >= 0, z0 = v0, z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0 >= 0, z0 = v0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0' >= 0, z0 = v0', z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z0 = 0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z0 = 0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0'', z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0'', v01 >= 0, z0 = v01
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0'', z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z0 = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1'', v0 >= 0, z0 = v0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0 >= 0, z0 = v0, z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0 >= 0, z0 = v0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, z0 = 0, v0 >= 0, z0 = v0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z0 = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1', v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0' >= 0, z0 = v0', z1' >= 0, z0'' >= 0, z0 = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z0 = 0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z0 = 0, v0' >= 0, z0 = v0'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0'', z1 >= 0, z0' >= 0, z0 = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0'', v01 >= 0, z0 = v01
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', v0'' >= 0, z0 = v0'', z0 = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z0 >= 0, z' = z0, v0 >= 0, z0 = v0, v0' >= 0, z0 = v0', z0 = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z1''), 1 + z1'', 1 + (1 + z041) + z2) + GE(1 + z041, 1 + z1'') :|: z' = 1 + (1 + z041) + z2, z = 1 + z1'', z041 >= 0, z2 >= 0, z1'' >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + z040, 1 + 0 + z2) + GE(0, 1 + z040) :|: z040 >= 0, z2 >= 0, z = 1 + z040, z' = 1 + 0 + z2
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z0, 1 + z1 + z2) + GE(z1, z0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z0) :|: z = z0, z0 >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z070, z123), 1 + z070, 1 + z123, z2) :|: z'' = z2, z123 >= 0, z' = 1 + z123, z070 >= 0, z2 >= 0, z = 1 + z070
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z0, 0, z2) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + z069, z2) :|: z'' = z2, z' = 1 + z069, z069 >= 0, z = 0, z2 >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z0, z1, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   average(z, z') -{ 0 }-> if(ge(z068, z122), 1 + z068, 1 + z122) :|: z = 1 + z068, z122 >= 0, z' = 1 + z122, z068 >= 0
   average(z, z') -{ 0 }-> if(2, z0, 0) :|: z = z0, z0 >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + z067) :|: z' = 1 + z067, z = 0, z067 >= 0
   average(z, z') -{ 0 }-> if(0, z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   average(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   high(z, z') -{ 0 }-> if_high(ge(z066, z121), 1 + z121, 1 + (1 + z066) + z2) :|: z121 >= 0, z = 1 + z121, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + z065, 1 + 0 + z2) :|: z = 1 + z065, z065 >= 0, z2 >= 0, z' = 1 + 0 + z2
   high(z, z') -{ 0 }-> if_high(0, z0, 1 + z1 + z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 0 }-> averIter(z0, z1, z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> averIter(z1, z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z2 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   low(z, z') -{ 0 }-> if_low(ge(z064, z120), 1 + z120, 1 + (1 + z064) + z2) :|: z120 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z = 1 + z120, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + z063, 1 + 0 + z2) :|: z063 >= 0, z = 1 + z063, z2 >= 0, z' = 1 + 0 + z2
   low(z, z') -{ 0 }-> if_low(0, z0, 1 + z1 + z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + plus(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


----------------------------------------

(121) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(122)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z', 1 + (1 + (1 + 0))) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z'', 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z3, 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, z') :|: z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0


----------------------------------------

(123) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { PLUS }
   { tail }
   { ge }
   { plus }
   { isempty }
   { head }
   { GE }
   { IFITER, AVERITER }
   { averIter, ifIter }
   { low, if_low }
   { IF_LOW, LOW }
   { IF }
   { if }
   { AVERAGE }
   { average }
   { HIGH, IF_HIGH }
   { high, if_high }
   { QUICKSORT, IFQUICK }

----------------------------------------

(124)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z', 1 + (1 + (1 + 0))) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z'', 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z3, 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, z') :|: z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PLUS}, {tail}, {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}

----------------------------------------

(125) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(126)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z', 1 + (1 + (1 + 0))) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z'', 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z3, 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, z') :|: z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PLUS}, {tail}, {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}

----------------------------------------

(127) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: PLUS
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(128)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z', 1 + (1 + (1 + 0))) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z'', 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z3, 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, z') :|: z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {PLUS}, {tail}, {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: ?, size: O(1) [0]

----------------------------------------

(129) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: PLUS
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(130)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z', 1 + (1 + (1 + 0))) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z'', 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + PLUS(z3, 1 + 0) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z - 1, z') :|: z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]

----------------------------------------

(131) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(132)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]

----------------------------------------

(133) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(134)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: ?, size: O(n^1) [z]

----------------------------------------

(135) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(136)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]

----------------------------------------

(137) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(138)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]

----------------------------------------

(139) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(140)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ge}, {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: ?, size: O(1) [2]

----------------------------------------

(141) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(142)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z043, z - 1), 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z041, z - 1), 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(ge(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> ge(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(ge(z066, z - 1), 1 + (z - 1), 1 + (1 + z066) + z2) :|: z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(ge(z064, z - 1), 1 + (z - 1), 1 + (1 + z064) + z2) :|: z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]

----------------------------------------

(143) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(144)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]

----------------------------------------

(145) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

----------------------------------------

(146)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {plus}, {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: ?, size: O(n^1) [z + z']

----------------------------------------

(147) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(148)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s :|: s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s' :|: s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) + s'' :|: s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z', 1 + (1 + (1 + 0))), plus(z'', 1 + 0), plus(z3, 1 + 0)) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']

----------------------------------------

(149) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(150)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']

----------------------------------------

(151) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(152)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isempty}, {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: ?, size: O(1) [2]

----------------------------------------

(153) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isempty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(154)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]

----------------------------------------

(155) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(156)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]

----------------------------------------

(157) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

----------------------------------------

(158)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: ?, size: O(n^1) [1 + z]

----------------------------------------

(159) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(160)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]

----------------------------------------

(161) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(162)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]

----------------------------------------

(163) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: GE
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(164)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {GE}, {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]
  GE: runtime: ?, size: O(1) [0]

----------------------------------------

(165) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: GE
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(166)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + GE(1 + (z - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + GE(z, 0) :|: z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + GE(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(0, z, z', z'') + GE(z, z') :|: z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ 1 }-> 1 + GE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + GE(1 + z043, 1 + (z - 1)) :|: s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + GE(1 + z041, 1 + (z - 1)) :|: s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + GE(z1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + GE(0, 1 + (z - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + GE(z1, z) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]
  GE: runtime: O(n^1) [z'], size: O(1) [0]

----------------------------------------

(167) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(168)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 + z' }-> 1 + s37 :|: s37 >= 0, s37 <= 0, z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + s26 :|: s26 >= 0, s26 <= 0, s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + s25 :|: s25 >= 0, s25 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(0, z, z', z'') + s27 :|: s27 >= 0, s27 <= 0, z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ z' }-> 1 + s36 :|: s36 >= 0, s36 <= 0, z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + s34 :|: s34 >= 0, s34 <= 0, s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + s32 :|: s32 >= 0, s32 <= 0, z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + s33 :|: s33 >= 0, s33 <= 0, z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + s35 :|: s35 >= 0, s35 <= 0, z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + s30 :|: s30 >= 0, s30 <= 0, s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + s28 :|: s28 >= 0, s28 <= 0, z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + s29 :|: s29 >= 0, s29 <= 0, z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + s31 :|: s31 >= 0, s31 <= 0, z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]
  GE: runtime: O(n^1) [z'], size: O(1) [0]

----------------------------------------

(169) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: IFITER
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: AVERITER
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(170)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 + z' }-> 1 + s37 :|: s37 >= 0, s37 <= 0, z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + s26 :|: s26 >= 0, s26 <= 0, s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + s25 :|: s25 >= 0, s25 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(0, z, z', z'') + s27 :|: s27 >= 0, s27 <= 0, z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ z' }-> 1 + s36 :|: s36 >= 0, s36 <= 0, z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + s34 :|: s34 >= 0, s34 <= 0, s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + s32 :|: s32 >= 0, s32 <= 0, z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + s33 :|: s33 >= 0, s33 <= 0, z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + s35 :|: s35 >= 0, s35 <= 0, z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + s30 :|: s30 >= 0, s30 <= 0, s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + s28 :|: s28 >= 0, s28 <= 0, z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + s29 :|: s29 >= 0, s29 <= 0, z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + s31 :|: s31 >= 0, s31 <= 0, z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]
  GE: runtime: O(n^1) [z'], size: O(1) [0]
  IFITER: runtime: ?, size: O(1) [0]
  AVERITER: runtime: ?, size: O(1) [1]

----------------------------------------

(171) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: IFITER
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(172)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 + z' }-> 1 + s37 :|: s37 >= 0, s37 <= 0, z' >= 0, z >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(s5, 1 + (z - 1), 1 + (z' - 1)) :|: s5 >= 0, s5 <= 2, z - 1 >= 0, z' - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(2, z, 0) :|: z >= 0, z' = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(1, 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0
   AVERAGE(z, z') -{ 1 }-> 1 + IF(0, z, z') :|: z' >= 0, z >= 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(s2, 1 + (z - 1), 1 + (z' - 1), z'') + s26 :|: s26 >= 0, s26 <= 0, s2 >= 0, s2 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(2, z, 0, z'') + s24 :|: s24 >= 0, s24 <= 0, z >= 0, z'' >= 0, z' = 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(1, 0, 1 + (z' - 1), z'') + s25 :|: s25 >= 0, s25 <= 0, z' - 1 >= 0, z = 0, z'' >= 0
   AVERITER(z, z', z'') -{ 1 + z' }-> 1 + IFITER(0, z, z', z'') + s27 :|: s27 >= 0, s27 <= 0, z' >= 0, z >= 0, z'' >= 0
   GE(z, z') -{ z' }-> 1 + s36 :|: s36 >= 0, s36 <= 0, z' - 1 >= 0, z - 1 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(s4, 1 + (z - 1), 1 + (1 + z043) + z2) + s34 :|: s34 >= 0, s34 <= 0, s4 >= 0, s4 <= 2, z' = 1 + (1 + z043) + z2, z - 1 >= 0, z043 >= 0, z2 >= 0
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(2, 0, 1 + z1 + z2) + s32 :|: s32 >= 0, s32 <= 0, z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + s33 :|: s33 >= 0, s33 <= 0, z - 1 >= 0, z' - 1 >= 0
   HIGH(z, z') -{ 1 + z }-> 1 + IF_HIGH(0, z, 1 + z1 + z2) + s35 :|: s35 >= 0, s35 <= 0, z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   IFITER(z, z', z'', z3) -{ 1 + z' }-> 1 + AVERITER(s11, s12, s13) + s :|: s11 >= 0, s11 <= z' + (1 + (1 + (1 + 0))), s12 >= 0, s12 <= z'' + (1 + 0), s13 >= 0, s13 <= z3 + (1 + 0), s >= 0, s <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z'' }-> 1 + AVERITER(s14, s15, s16) + s' :|: s14 >= 0, s14 <= z' + (1 + (1 + (1 + 0))), s15 >= 0, s15 <= z'' + (1 + 0), s16 >= 0, s16 <= z3 + (1 + 0), s' >= 0, s' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFITER(z, z', z'', z3) -{ 1 + z3 }-> 1 + AVERITER(s17, s18, s19) + s'' :|: s17 >= 0, s17 <= z' + (1 + (1 + (1 + 0))), s18 >= 0, s18 <= z'' + (1 + 0), s19 >= 0, s19 <= z3 + (1 + 0), s'' >= 0, s'' <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1')) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', z1'')) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(z0', 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(1, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1)) + LOW(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, z1')) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(0, 0)) + LOW(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1')) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', z11) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1'', z11 >= 0, z02 >= 0, z' = 1 + z02 + z11
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', z1'')) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(z0'', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(z0', 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(1, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1)) + HIGH(0, 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', z1'') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1', z1'' >= 0, z01 >= 0, z' = 1 + z01 + z1''
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, z1')) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', z1') :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1, z1' >= 0, z0'' >= 0, z' = 1 + z0'' + z1'
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(z0', 0) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(1, 0) :|: z = 1, z' >= 0, z' = 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, z1) :|: z = 1, z' >= 0, z1 >= 0, z0' >= 0, z' = 1 + z0' + z1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(0, 0)) + HIGH(0, 0) :|: z = 1, z' >= 0, z' = 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(s3, 1 + (z - 1), 1 + (1 + z041) + z2) + s30 :|: s30 >= 0, s30 <= 0, s3 >= 0, s3 <= 2, z' = 1 + (1 + z041) + z2, z041 >= 0, z2 >= 0, z - 1 >= 0
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(2, 0, 1 + z1 + z2) + s28 :|: s28 >= 0, s28 <= 0, z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(1, 1 + (z - 1), 1 + 0 + (z' - 1)) + s29 :|: s29 >= 0, s29 <= 0, z - 1 >= 0, z' - 1 >= 0
   LOW(z, z') -{ 1 + z }-> 1 + IF_LOW(0, z, 1 + z1 + z2) + s31 :|: s31 >= 0, s31 <= 0, z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   PLUS(z, z') -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(2, 0) :|: z = 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(1, 1 + z044 + z12) :|: z044 >= 0, z12 >= 0, z = 1 + z044 + z12
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(0, z) :|: z >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(s10, 1 + (z - 1), 1 + (z' - 1), z'') :|: s10 >= 0, s10 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(2, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   averIter(z, z', z'') -{ 0 }-> ifIter(1, 0, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(0, z, z', z'') :|: z' >= 0, z >= 0, z'' >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   average(z, z') -{ 0 }-> if(s9, 1 + (z - 1), 1 + (z' - 1)) :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   average(z, z') -{ 0 }-> if(2, z, 0) :|: z >= 0, z' = 0
   average(z, z') -{ 0 }-> if(1, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0
   average(z, z') -{ 0 }-> if(0, z, z') :|: z' >= 0, z >= 0
   average(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ge(z, z') -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z - 1 >= 0
   ge(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: z >= 0
   high(z, z') -{ 0 }-> if_high(s8, 1 + (z - 1), 1 + (1 + z066) + z2) :|: s8 >= 0, s8 <= 2, z - 1 >= 0, z' = 1 + (1 + z066) + z2, z066 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   high(z, z') -{ 0 }-> if_high(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   high(z, z') -{ 0 }-> if_high(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z', z'', z') :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 0 }-> averIter(z'', z', z'') :|: z = 2, z'' >= 0, z' >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(s21, s22, s23) :|: s21 >= 0, s21 <= z' + (1 + (1 + (1 + 0))), s22 >= 0, s22 <= z'' + (1 + 0), s23 >= 0, s23 <= z3 + (1 + 0), z'' >= 0, z = 1, z' >= 0, z3 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z', z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z', z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z' >= 0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: z >= 0
   low(z, z') -{ 0 }-> if_low(s7, 1 + (z - 1), 1 + (1 + z064) + z2) :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z064 >= 0, z' = 1 + (1 + z064) + z2, z2 >= 0
   low(z, z') -{ 0 }-> if_low(2, 0, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 0, z2 >= 0
   low(z, z') -{ 0 }-> if_low(1, 1 + (z - 1), 1 + 0 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   low(z, z') -{ 0 }-> if_low(0, z, 1 + z1 + z2) :|: z1 >= 0, z' = 1 + z1 + z2, z >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s20 :|: s20 >= 0, s20 <= z - 1 + z', z' >= 0, z - 1 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFITER,AVERITER}, {averIter,ifIter}, {low,if_low}, {IF_LOW,LOW}, {IF}, {if}, {AVERAGE}, {average}, {HIGH,IF_HIGH}, {high,if_high}, {QUICKSORT,IFQUICK}
Previous analysis results are:
  PLUS: runtime: O(n^1) [z], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  ge: runtime: O(1) [0], size: O(1) [2]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  isempty: runtime: O(1) [0], size: O(1) [2]
  head: runtime: O(1) [0], size: O(n^1) [1 + z]
  GE: runtime: O(n^1) [z'], size: O(1) [0]
  IFITER: runtime: INF, size: O(1) [0]
  AVERITER: runtime: ?, size: O(1) [1]

----------------------------------------

(173) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   low(v0, v1) -> null_low [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   high(v0, v1) -> null_high [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   average(v0, v1) -> null_average [0]
   if(v0, v1, v2) -> null_if [0]
   averIter(v0, v1, v2) -> null_averIter [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]
   IF(v0, v1, v2) -> null_IF [0]
   IFITER(v0, v1, v2, v3) -> null_IFITER [0]
   LOW(v0, v1) -> null_LOW [0]
   IF_LOW(v0, v1, v2) -> null_IF_LOW [0]
   HIGH(v0, v1) -> null_HIGH [0]
   IF_HIGH(v0, v1, v2) -> null_IF_HIGH [0]
   PLUS(v0, v1) -> null_PLUS [0]
   GE(v0, v1) -> null_GE [0]
   IFQUICK(v0, v1) -> null_IFQUICK [0]

And the following fresh constants:    null_ge, null_plus, null_isempty, null_low, null_head, null_tail, null_if_low, null_high, null_if_high, null_average, null_if, null_averIter, null_ifIter, null_IF, null_IFITER, null_LOW, null_IF_LOW, null_HIGH, null_IF_HIGH, null_PLUS, null_GE, null_IFQUICK, const, const1, const2

----------------------------------------

(174)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1)) [1]
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0)) [1]
   AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1)) [1]
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0))))) [1]
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0))) [1]
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0))) [1]
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2)) [1]
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2)) [1]
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) [1]
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1)) [1]
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2)) [1]
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1)) [1]
   GE(s(z0), s(z1)) -> c37(GE(z0, z1)) [1]
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0)) [1]
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) [1]
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1)) [1]
   AVERAGE(z0, z1) -> c5(GE(z0, z1)) [1]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   plus(0, z0) -> z0 [0]
   plus(s(z0), z1) -> s(plus(z0, z1)) [0]
   isempty(nil) -> true [0]
   isempty(cons(z0, z1)) -> false [0]
   low(z0, nil) -> nil [0]
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2)) [0]
   head(nil) -> error [0]
   head(cons(z0, z1)) -> z0 [0]
   tail(nil) -> nil [0]
   tail(cons(z0, z1)) -> z1 [0]
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2)) [0]
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2) [0]
   high(z0, nil) -> nil [0]
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2)) [0]
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2) [0]
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2)) [0]
   average(z0, z1) -> if(ge(z0, z1), z0, z1) [0]
   if(true, z0, z1) -> averIter(z1, z0, z1) [0]
   if(false, z0, z1) -> averIter(z0, z1, z0) [0]
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2) [0]
   ifIter(true, z0, z1, z2) -> z2 [0]
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))) [0]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   isempty(v0) -> null_isempty [0]
   low(v0, v1) -> null_low [0]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   if_low(v0, v1, v2) -> null_if_low [0]
   high(v0, v1) -> null_high [0]
   if_high(v0, v1, v2) -> null_if_high [0]
   average(v0, v1) -> null_average [0]
   if(v0, v1, v2) -> null_if [0]
   averIter(v0, v1, v2) -> null_averIter [0]
   ifIter(v0, v1, v2, v3) -> null_ifIter [0]
   IF(v0, v1, v2) -> null_IF [0]
   IFITER(v0, v1, v2, v3) -> null_IFITER [0]
   LOW(v0, v1) -> null_LOW [0]
   IF_LOW(v0, v1, v2) -> null_IF_LOW [0]
   HIGH(v0, v1) -> null_HIGH [0]
   IF_HIGH(v0, v1, v2) -> null_IF_HIGH [0]
   PLUS(v0, v1) -> null_PLUS [0]
   GE(v0, v1) -> null_GE [0]
   IFQUICK(v0, v1) -> null_IFQUICK [0]

The TRS has the following type information:
IF :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c2:c3:c37:null_IF:null_GE
true :: true:false:null_ge:null_isempty
c2 :: c4 -> c2:c3:c37:null_IF:null_GE
AVERITER :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c4
false :: true:false:null_ge:null_isempty
c3 :: c4 -> c2:c3:c37:null_IF:null_GE
c4 :: c6:c7:c8:null_IFITER -> c2:c3:c37:null_IF:null_GE -> c4
IFITER :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c6:c7:c8:null_IFITER
ge :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> true:false:null_ge:null_isempty
GE :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c2:c3:c37:null_IF:null_GE
c6 :: c4 -> c28:null_PLUS -> c6:c7:c8:null_IFITER
plus :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
s :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
0 :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
PLUS :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c28:null_PLUS
c7 :: c4 -> c28:null_PLUS -> c6:c7:c8:null_IFITER
c8 :: c4 -> c28:null_PLUS -> c6:c7:c8:null_IFITER
LOW :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> c12:null_LOW
cons :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
c12 :: c13:c14:null_IF_LOW -> c2:c3:c37:null_IF:null_GE -> c12:null_LOW
IF_LOW :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> c13:c14:null_IF_LOW
c13 :: c12:null_LOW -> c13:c14:null_IF_LOW
c14 :: c12:null_LOW -> c13:c14:null_IF_LOW
HIGH :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> c16:null_HIGH
c16 :: c17:c18:c19:null_IF_HIGH -> c2:c3:c37:null_IF:null_GE -> c16:null_HIGH
IF_HIGH :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> c17:c18:c19:null_IF_HIGH
c17 :: c16:null_HIGH -> c17:c18:c19:null_IF_HIGH
c18 :: c5 -> c17:c18:c19:null_IF_HIGH
AVERAGE :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> c5
c19 :: c16:null_HIGH -> c17:c18:c19:null_IF_HIGH
c28 :: c28:null_PLUS -> c28:null_PLUS
c37 :: c2:c3:c37:null_IF:null_GE -> c2:c3:c37:null_IF:null_GE
QUICKSORT :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> c20
c20 :: c22:c23:c25:c26:null_IFQUICK -> c20
IFQUICK :: true:false:null_ge:null_isempty -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> c22:c23:c25:c26:null_IFQUICK
isempty :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> true:false:null_ge:null_isempty
c22 :: c20 -> c12:null_LOW -> c22:c23:c25:c26:null_IFQUICK
low :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
head :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
tail :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
c23 :: c20 -> c12:null_LOW -> c22:c23:c25:c26:null_IFQUICK
c25 :: c20 -> c16:null_HIGH -> c22:c23:c25:c26:null_IFQUICK
high :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
c26 :: c20 -> c16:null_HIGH -> c22:c23:c25:c26:null_IFQUICK
c5 :: c2:c3:c37:null_IF:null_GE -> c5
nil :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
if_low :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
error :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
if_high :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high -> cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
average :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
if :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
averIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
ifIter :: true:false:null_ge:null_isempty -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter -> 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_ge :: true:false:null_ge:null_isempty
null_plus :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_isempty :: true:false:null_ge:null_isempty
null_low :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
null_head :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_tail :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
null_if_low :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
null_high :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
null_if_high :: cons:nil:null_low:null_tail:null_if_low:null_high:null_if_high
null_average :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_if :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_averIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_ifIter :: 0:s:error:null_plus:null_head:null_average:null_if:null_averIter:null_ifIter
null_IF :: c2:c3:c37:null_IF:null_GE
null_IFITER :: c6:c7:c8:null_IFITER
null_LOW :: c12:null_LOW
null_IF_LOW :: c13:c14:null_IF_LOW
null_HIGH :: c16:null_HIGH
null_IF_HIGH :: c17:c18:c19:null_IF_HIGH
null_PLUS :: c28:null_PLUS
null_GE :: c2:c3:c37:null_IF:null_GE
null_IFQUICK :: c22:c23:c25:c26:null_IFQUICK
const :: c4
const1 :: c5
const2 :: c20

Rewrite Strategy: INNERMOST
----------------------------------------

(175) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   false => 1
   0 => 0
   nil => 0
   error => 1
   null_ge => 0
   null_plus => 0
   null_isempty => 0
   null_low => 0
   null_head => 0
   null_tail => 0
   null_if_low => 0
   null_high => 0
   null_if_high => 0
   null_average => 0
   null_if => 0
   null_averIter => 0
   null_ifIter => 0
   null_IF => 0
   null_IFITER => 0
   null_LOW => 0
   null_IF_LOW => 0
   null_HIGH => 0
   null_IF_HIGH => 0
   null_PLUS => 0
   null_GE => 0
   null_IFQUICK => 0
   const => 0
   const1 => 0
   const2 => 0

----------------------------------------

(176)
Obligation:
Complexity RNTS consisting of the following rules:

   AVERAGE(z, z') -{ 1 }-> 1 + IF(ge(z0, z1), z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   AVERITER(z, z', z'') -{ 1 }-> 1 + IFITER(ge(z0, z1), z0, z1, z2) + GE(z0, z1) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   GE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   HIGH(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   HIGH(z, z') -{ 1 }-> 1 + IF_HIGH(ge(z1, z0), z0, 1 + z1 + z2) + GE(z1, z0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   IF(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z0, z1, z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   IF(z, z', z'') -{ 1 }-> 1 + AVERITER(z1, z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   IFITER(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z0, 1 + (1 + (1 + 0))) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z1, 1 + 0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFITER(z, z', z'', z3) -{ 1 }-> 1 + AVERITER(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) + PLUS(z2, 1 + 0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IFQUICK(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(low(head(z0), tail(z0))) + LOW(head(z0), tail(z0)) :|: z = 1, z0 >= 0, z' = z0
   IFQUICK(z, z') -{ 1 }-> 1 + QUICKSORT(high(head(z0), tail(z0))) + HIGH(head(z0), tail(z0)) :|: z = 1, z0 >= 0, z' = z0
   IF_HIGH(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + HIGH(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_HIGH(z, z', z'') -{ 1 }-> 1 + AVERAGE(z1, z1) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_LOW(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   IF_LOW(z, z', z'') -{ 1 }-> 1 + LOW(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   LOW(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   LOW(z, z') -{ 1 }-> 1 + IF_LOW(ge(z1, z0), z0, 1 + z1 + z2) + GE(z1, z0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   PLUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   PLUS(z, z') -{ 1 }-> 1 + PLUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   QUICKSORT(z) -{ 1 }-> 1 + IFQUICK(isempty(z0), z0) :|: z = z0, z0 >= 0
   averIter(z, z', z'') -{ 0 }-> ifIter(ge(z0, z1), z0, z1, z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   averIter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   average(z, z') -{ 0 }-> if(ge(z0, z1), z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   average(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 1 :|: z = 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   high(z, z') -{ 0 }-> if_high(ge(z1, z0), z0, 1 + z1 + z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   high(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   high(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 0 }-> averIter(z0, z1, z0) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> averIter(z1, z0, z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   ifIter(z, z', z'', z3) -{ 0 }-> z2 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   ifIter(z, z', z'', z3) -{ 0 }-> averIter(plus(z0, 1 + (1 + (1 + 0))), plus(z1, 1 + 0), plus(z2, 1 + 0)) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   ifIter(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   if_high(z, z', z'') -{ 0 }-> high(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_high(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_high(z, z', z'') -{ 0 }-> 1 + average(z1, z1) + high(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> low(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   if_low(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_low(z, z', z'') -{ 0 }-> 1 + z1 + low(z0, z2) :|: z1 >= 0, z = 1, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   isempty(z) -{ 0 }-> 2 :|: z = 0
   isempty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   isempty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   low(z, z') -{ 0 }-> if_low(ge(z1, z0), z0, 1 + z1 + z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   low(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   low(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + plus(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace AVERITER(z0, z1, z2) -> c4(IFITER(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by 
   AVERITER(z0, 0, x2) -> c4(IFITER(true, z0, 0, x2), GE(z0, 0))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2), GE(0, s(z0)))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))

----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(z0, 0, x2) -> c4(IFITER(true, z0, 0, x2), GE(z0, 0))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2), GE(0, s(z0)))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(z0, 0, x2) -> c4(IFITER(true, z0, 0, x2), GE(z0, 0))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2), GE(0, s(z0)))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2


----------------------------------------

(179) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   AVERITER(z0, 0, x2) -> c4(IFITER(true, z0, 0, x2), GE(z0, 0))

----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2), GE(0, s(z0)))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2), GE(0, s(z0)))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2


----------------------------------------

(181) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0)))))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c6_2, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1


----------------------------------------

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, z0, z1, z2) -> c6(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z0, s(s(s(0))))) by 
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))

----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2


----------------------------------------

(185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1


----------------------------------------

(187) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(PLUS(x0, s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(PLUS(x0, s(s(s(0)))))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1


----------------------------------------

(189) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, x0, 0, x2) -> c(PLUS(x0, s(s(s(0)))))

----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1


----------------------------------------

(191) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))

----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0)))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c7_2, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1


----------------------------------------

(193) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, z0, z1, z2) -> c7(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z1, s(0))) by 
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))

----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2


----------------------------------------

(195) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1


----------------------------------------

(197) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))

----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0)))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IFITER_4, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3

Compound Symbols:   c2_1, c3_1, c8_2, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1


----------------------------------------

(199) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, z0, z1, z2) -> c8(AVERITER(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0))), PLUS(z2, s(0))) by 
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, 0, x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))

----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, 0, x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, 0, x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4

Compound Symbols:   c2_1, c3_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2


----------------------------------------

(201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, 0, x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, 0, x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4

Compound Symbols:   c2_1, c3_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1


----------------------------------------

(203) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(PLUS(x2, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(PLUS(x2, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4

Compound Symbols:   c2_1, c3_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1


----------------------------------------

(205) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, x0, 0, x2) -> c1(PLUS(x2, s(0)))

----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4

Compound Symbols:   c2_1, c3_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1


----------------------------------------

(207) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))

----------------------------------------

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, LOW_2, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4

Compound Symbols:   c2_1, c3_1, c12_2, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1


----------------------------------------

(209) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOW(z0, cons(z1, z2)) -> c12(IF_LOW(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) by 
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)), GE(z0, 0))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)), GE(0, s(z0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))

----------------------------------------

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)), GE(z0, 0))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)), GE(0, s(z0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)), GE(z0, 0))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)), GE(0, s(z0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2


----------------------------------------

(211) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IF_LOW_3, HIGH_2, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c16_2, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1


----------------------------------------

(213) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HIGH(z0, cons(z1, z2)) -> c16(IF_HIGH(ge(z1, z0), z0, cons(z1, z2)), GE(z1, z0)) by 
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)), GE(z0, 0))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)), GE(0, s(z0)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))

----------------------------------------

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)), GE(z0, 0))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)), GE(0, s(z0)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)), GE(z0, 0))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)), GE(0, s(z0)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2


----------------------------------------

(215) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, QUICKSORT_1, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c20_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1


----------------------------------------

(217) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace QUICKSORT(z0) -> c20(IFQUICK(isempty(z0), z0)) by 
   QUICKSORT(nil) -> c20(IFQUICK(true, nil))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))

----------------------------------------

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(nil) -> c20(IFQUICK(true, nil))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(nil) -> c20(IFQUICK(true, nil))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1


----------------------------------------

(219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   QUICKSORT(nil) -> c20(IFQUICK(true, nil))

----------------------------------------

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   ge_2, plus_2, isempty_1, low_2, head_1, tail_1, if_low_3, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1


----------------------------------------

(221) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false

----------------------------------------

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c22_2, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1


----------------------------------------

(223) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, z0) -> c22(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) by 
   IFQUICK(false, nil) -> c22(QUICKSORT(low(head(nil), nil)), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c22(QUICKSORT(low(error, tail(nil))), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c22(QUICKSORT(low(head(nil), nil)), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c22(QUICKSORT(low(error, tail(nil))), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c22(QUICKSORT(low(head(nil), nil)), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c22(QUICKSORT(low(error, tail(nil))), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2


----------------------------------------

(225) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1


----------------------------------------

(227) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))

----------------------------------------

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1


----------------------------------------

(229) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_1
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c23_2, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1


----------------------------------------

(231) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, z0) -> c23(QUICKSORT(low(head(z0), tail(z0))), LOW(head(z0), tail(z0))) by 
   IFQUICK(false, nil) -> c23(QUICKSORT(low(head(nil), nil)), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c23(QUICKSORT(low(error, tail(nil))), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c23(QUICKSORT(low(head(nil), nil)), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c23(QUICKSORT(low(error, tail(nil))), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c23(QUICKSORT(low(head(nil), nil)), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c23(QUICKSORT(low(error, tail(nil))), LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2


----------------------------------------

(233) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(234)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1


----------------------------------------

(235) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))

----------------------------------------

(236)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1


----------------------------------------

(237) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_1
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(238)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c25_2, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1


----------------------------------------

(239) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, z0) -> c25(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) by 
   IFQUICK(false, nil) -> c25(QUICKSORT(high(head(nil), nil)), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c25(QUICKSORT(high(error, tail(nil))), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(240)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c25(QUICKSORT(high(head(nil), nil)), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c25(QUICKSORT(high(error, tail(nil))), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c25(QUICKSORT(high(head(nil), nil)), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c25(QUICKSORT(high(error, tail(nil))), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2


----------------------------------------

(241) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(242)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1


----------------------------------------

(243) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))

----------------------------------------

(244)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1


----------------------------------------

(245) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_1
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(246)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0)))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, IFQUICK_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c26_2, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1


----------------------------------------

(247) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, z0) -> c26(QUICKSORT(high(head(z0), tail(z0))), HIGH(head(z0), tail(z0))) by 
   IFQUICK(false, nil) -> c26(QUICKSORT(high(head(nil), nil)), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c26(QUICKSORT(high(error, tail(nil))), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(248)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, nil) -> c26(QUICKSORT(high(head(nil), nil)), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c26(QUICKSORT(high(error, tail(nil))), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c26(QUICKSORT(high(head(nil), nil)), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c26(QUICKSORT(high(error, tail(nil))), HIGH(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2


----------------------------------------

(249) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(250)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1


----------------------------------------

(251) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))

----------------------------------------

(252)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1


----------------------------------------

(253) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_1
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(254)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1


----------------------------------------

(255) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace AVERAGE(z0, z1) -> c5(IF(ge(z0, z1), z0, z1)) by 
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(0, s(z0)) -> c5(IF(false, 0, s(z0)))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))

----------------------------------------

(256)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(0, s(z0)) -> c5(IF(false, 0, s(z0)))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(0, s(z0)) -> c5(IF(false, 0, s(z0)))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1


----------------------------------------

(257) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   AVERAGE(0, s(z0)) -> c5(IF(false, 0, s(z0)))

----------------------------------------

(258)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1)))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_2, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1


----------------------------------------

(259) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace AVERITER(s(z0), s(z1), x2) -> c4(IFITER(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by 
   AVERITER(s(z0), s(0), x2) -> c4(IFITER(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))

----------------------------------------

(260)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(z0), s(0), x2) -> c4(IFITER(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(z0), s(0), x2) -> c4(IFITER(true, s(z0), s(0), x2), GE(s(z0), s(0)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2


----------------------------------------

(261) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(262)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2


----------------------------------------

(263) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, x1, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x0, s(s(s(0))))) by 
   IFITER(false, x0, 0, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))

----------------------------------------

(264)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, 0, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, 0, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2


----------------------------------------

(265) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(266)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, 0, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, 0, 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2


----------------------------------------

(267) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(268)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c21(PLUS(x0, s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c21(PLUS(x0, s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1


----------------------------------------

(269) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, x0, 0, 0) -> c21(PLUS(x0, s(s(s(0)))))

----------------------------------------

(270)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1


----------------------------------------

(271) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))

----------------------------------------

(272)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1


----------------------------------------

(273) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, x1, s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0))))) by 
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))

----------------------------------------

(274)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1


----------------------------------------

(275) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(276)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1


----------------------------------------

(277) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(278)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(PLUS(x0, s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(PLUS(x0, s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1, c24_1


----------------------------------------

(279) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, x0, 0, s(x2)) -> c24(PLUS(x0, s(s(s(0)))))

----------------------------------------

(280)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1, c24_1


----------------------------------------

(281) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))

----------------------------------------

(282)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1, c24_1


----------------------------------------

(283) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, s(z0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x0, s(s(s(0))))) by 
   IFITER(false, x0, s(x1), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(x1), s(z0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(plus(z0, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))

----------------------------------------

(284)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1, c24_1


----------------------------------------

(285) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(286)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c21_1, c24_1


----------------------------------------

(287) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, s(z0), x1, x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(z0), s(s(s(0))))) by 
   IFITER(false, s(x0), x1, 0) -> c6(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), x1, s(z0)) -> c6(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c6(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), s(z0), x2) -> c6(AVERITER(s(plus(x0, s(s(s(0))))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))

----------------------------------------

(288)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(x0), 0, x2) -> c6(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(x0), 0, x2) -> c6(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c21_1, c24_1


----------------------------------------

(289) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(290)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, s(x0), 0, x2) -> c27(PLUS(s(x0), s(s(s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, s(x0), 0, x2) -> c27(PLUS(s(x0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c21_1, c24_1, c27_1


----------------------------------------

(291) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, s(x0), 0, x2) -> c27(PLUS(s(x0), s(s(s(0)))))

----------------------------------------

(292)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c21_1, c24_1, c27_1


----------------------------------------

(293) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))

----------------------------------------

(294)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c21_1, c24_1, c27_1


----------------------------------------

(295) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, 0, x1, x2) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0)))) by 
   IFITER(false, 0, x0, 0) -> c6(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, x0, s(z0)) -> c6(AVERITER(s(s(s(0))), plus(x0, s(0)), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, 0, s(z0), x1) -> c6(AVERITER(s(s(s(0))), s(plus(z0, s(0))), plus(x1, s(0))))

----------------------------------------

(296)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1


----------------------------------------

(297) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))

----------------------------------------

(298)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1


----------------------------------------

(299) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0)))) by 
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))

----------------------------------------

(300)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(301) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, x1, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)), PLUS(x1, s(0))) by 
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))

----------------------------------------

(302)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(303) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(304)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(305) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))

----------------------------------------

(306)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(307) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, x1, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0))) by 
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))

----------------------------------------

(308)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(0, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(309) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(310)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(311) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))

----------------------------------------

(312)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(313) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, s(z0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0))) by 
   IFITER(false, x0, s(x1), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(plus(z0, s(0)))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))

----------------------------------------

(314)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(315) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, 0, x1, x2) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0))) by 
   IFITER(false, 0, x0, 0) -> c7(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)), PLUS(x0, s(0)))
   IFITER(false, 0, x0, s(z0)) -> c7(AVERITER(s(s(s(0))), plus(x0, s(0)), s(plus(z0, s(0)))), PLUS(x0, s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(0, s(0)))
   IFITER(false, 0, s(z0), x1) -> c7(AVERITER(s(s(s(0))), s(plus(z0, s(0))), plus(x1, s(0))), PLUS(s(z0), s(0)))

----------------------------------------

(316)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(0, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(0, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(317) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(318)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(319) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))

----------------------------------------

(320)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_2, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1


----------------------------------------

(321) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, s(z0), x1, x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0))) by 
   IFITER(false, s(x0), x1, 0) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(x0), x1, s(z0)) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(0, s(0)))
   IFITER(false, s(x0), s(z0), x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(s(z0), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))

----------------------------------------

(322)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(0, s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(0, s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2


----------------------------------------

(323) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(324)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2


----------------------------------------

(325) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))

----------------------------------------

(326)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c7_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2


----------------------------------------

(327) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0)))) by 
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))

----------------------------------------

(328)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1


----------------------------------------

(329) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, x1, s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0))) by 
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))

----------------------------------------

(330)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1


----------------------------------------

(331) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(332)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c29(PLUS(s(x2), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c29(PLUS(s(x2), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(333) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, x0, 0, s(x2)) -> c29(PLUS(s(x2), s(0)))

----------------------------------------

(334)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(335) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))

----------------------------------------

(336)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(337) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, s(z0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0))) by 
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, s(x1), s(z0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))

----------------------------------------

(338)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)), PLUS(0, s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(339) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(340)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(341) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, 0, x1, x2) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0))) by 
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, 0, x0, s(z0)) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, 0, x1) -> c8(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(x1, s(0)))
   IFITER(false, 0, s(z0), x1) -> c8(AVERITER(s(s(s(0))), s(plus(z0, s(0))), plus(x1, s(0))), PLUS(x1, s(0)))

----------------------------------------

(342)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, 0, 0, x1) -> c8(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(x1, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, 0, 0, x1) -> c8(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(343) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(344)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, 0, x1) -> c8(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(x1, s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, 0, x1) -> c8(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))), PLUS(x1, s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1


----------------------------------------

(345) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(346)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, 0, 0, x1) -> c30(PLUS(x1, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, 0, 0, x1) -> c30(PLUS(x1, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1, c30_1


----------------------------------------

(347) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, 0, 0, x1) -> c30(PLUS(x1, s(0)))

----------------------------------------

(348)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1, c30_1


----------------------------------------

(349) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))

----------------------------------------

(350)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_2, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c29_1, c30_1


----------------------------------------

(351) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, s(z0), x1, x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0))) by 
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, s(x0), x1, s(z0)) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(plus(z0, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, s(x0), 0, x2) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), s(z0), x2) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), s(plus(z0, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))

----------------------------------------

(352)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, s(x0), 0, x2) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(0, s(0)))
   IFITER(false, s(x0), 0, x2) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c29_1, c30_1


----------------------------------------

(353) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(354)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c29_1, c30_1


----------------------------------------

(355) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(356)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(PLUS(x2, s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(PLUS(x2, s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c29_1, c30_1, c31_1


----------------------------------------

(357) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IFITER(false, s(x0), 0, x2) -> c31(PLUS(x2, s(0)))

----------------------------------------

(358)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c29_1, c30_1, c31_1


----------------------------------------

(359) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))

----------------------------------------

(360)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0)))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c8_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c29_1, c30_1, c31_1


----------------------------------------

(361) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, x1, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(0))) by 
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)))
   IFITER(false, 0, x1, 0) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, s(z0), x1, 0) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)))

----------------------------------------

(362)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1


----------------------------------------

(363) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))

----------------------------------------

(364)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, IFITER_4, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c1_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1


----------------------------------------

(365) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0)))) by 
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))

----------------------------------------

(366)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_2, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1


----------------------------------------

(367) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOW(s(z1), cons(s(z0), x2)) -> c12(IF_LOW(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1))) by 
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))

----------------------------------------

(368)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_2, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2


----------------------------------------

(369) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HIGH(s(z1), cons(s(z0), x2)) -> c16(IF_HIGH(ge(z0, z1), s(z1), cons(s(z0), x2)), GE(s(z0), s(z1))) by 
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))

----------------------------------------

(370)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2


----------------------------------------

(371) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1)))) by 
   IFQUICK(false, cons(x0, nil)) -> c22(QUICKSORT(nil), LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(372)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, nil)) -> c22(QUICKSORT(nil), LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2


----------------------------------------

(373) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(374)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c22_2, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_1


----------------------------------------

(375) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1)))) by 
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(376)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(377) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c27(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = [1]
   POL(plus(x_1, x_2)) = x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(378)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(379) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil))) by 
   IFQUICK(false, nil) -> c9(QUICKSORT(nil))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))

----------------------------------------

(380)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(nil))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFITER(false, x0, 0, x2) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), plus(x2, s(0))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(381) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFQUICK(false, nil) -> c9(QUICKSORT(nil))

----------------------------------------

(382)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(383) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil))) by 
   IFQUICK(false, nil) -> c9(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c9(LOW(error, tail(nil)))

----------------------------------------

(384)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c9(LOW(error, tail(nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(385) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   IFQUICK(false, nil) -> c9(LOW(error, tail(nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), nil))

----------------------------------------

(386)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(387) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil)))) by 
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))

----------------------------------------

(388)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c9_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1


----------------------------------------

(389) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil))) by 
   IFQUICK(false, nil) -> c9(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c9(LOW(error, tail(nil)))

----------------------------------------

(390)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c9(LOW(error, tail(nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c9(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1


----------------------------------------

(391) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   IFQUICK(false, nil) -> c9(LOW(error, tail(nil)))
   IFQUICK(false, nil) -> c9(LOW(head(nil), nil))

----------------------------------------

(392)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1


----------------------------------------

(393) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1)))) by 
   IFQUICK(false, cons(x0, nil)) -> c23(QUICKSORT(nil), LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(394)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, nil)) -> c23(QUICKSORT(nil), LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1


----------------------------------------

(395) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(396)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c23_2, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_1


----------------------------------------

(397) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, tail(cons(z0, z1)))), LOW(head(cons(z0, z1)), tail(cons(z0, z1)))) by 
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))

----------------------------------------

(398)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(399) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = 0
   POL(LOW(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c27(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(400)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(401) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
We considered the (Usable) Rules:
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   tail(cons(z0, z1)) -> z1
   low(z0, nil) -> nil
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   high(z0, nil) -> nil
   tail(nil) -> nil
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
And the Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(AVERAGE(x_1, x_2)) = 0
   POL(AVERITER(x_1, x_2, x_3)) = 0
   POL(GE(x_1, x_2)) = 0
   POL(HIGH(x_1, x_2)) = 0
   POL(IF(x_1, x_2, x_3)) = 0
   POL(IFITER(x_1, x_2, x_3, x_4)) = 0
   POL(IFQUICK(x_1, x_2)) = x_1 + x_2
   POL(IF_HIGH(x_1, x_2, x_3)) = 0
   POL(IF_LOW(x_1, x_2, x_3)) = [1]
   POL(LOW(x_1, x_2)) = [1]
   POL(PLUS(x_1, x_2)) = x_2
   POL(QUICKSORT(x_1)) = x_1
   POL(averIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(average(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c16(x_1, x_2)) = x_1 + x_2
   POL(c17(x_1)) = x_1
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c23(x_1)) = x_1
   POL(c23(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1)) = x_1
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c26(x_1, x_2)) = x_1 + x_2
   POL(c27(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c29(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c37(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(head(x_1)) = 0
   POL(high(x_1, x_2)) = x_2
   POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(ifIter(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(if_high(x_1, x_2, x_3)) = x_3
   POL(if_low(x_1, x_2, x_3)) = x_3
   POL(low(x_1, x_2)) = x_2
   POL(nil) = [1]
   POL(plus(x_1, x_2)) = x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

----------------------------------------

(402)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(403) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil))) by 
   IFQUICK(false, nil) -> c10(QUICKSORT(nil))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))

----------------------------------------

(404)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(nil))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(QUICKSORT(low(head(nil), nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(head(cons(z0, z1)), z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(405) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IFQUICK(false, nil) -> c10(QUICKSORT(nil))

----------------------------------------

(406)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(407) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil))) by 
   IFQUICK(false, nil) -> c10(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c10(LOW(error, tail(nil)))

----------------------------------------

(408)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c10(LOW(error, tail(nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(409) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   IFQUICK(false, nil) -> c10(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c10(LOW(error, tail(nil)))

----------------------------------------

(410)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(411) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil)))) by 
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))

----------------------------------------

(412)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c10_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1


----------------------------------------

(413) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil))) by 
   IFQUICK(false, nil) -> c10(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c10(LOW(error, tail(nil)))

----------------------------------------

(414)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), nil))
   IFQUICK(false, nil) -> c10(LOW(error, tail(nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c10(LOW(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, tail(nil))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1, c10_1


----------------------------------------

(415) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   IFQUICK(false, nil) -> c10(LOW(error, tail(nil)))
   IFQUICK(false, nil) -> c10(LOW(head(nil), nil))

----------------------------------------

(416)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   low(z0, nil) -> nil
   low(z0, cons(z1, z2)) -> if_low(ge(z1, z0), z0, cons(z1, z2))
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   if_low(false, z0, cons(z1, z2)) -> cons(z1, low(z0, z2))
   if_low(true, z0, cons(z1, z2)) -> low(z0, z2)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   high(z0, nil) -> nil
   high(z0, cons(z1, z2)) -> if_high(ge(z1, z0), z0, cons(z1, z2))
   if_high(false, z0, cons(z1, z2)) -> high(z0, z2)
   if_high(true, z0, cons(z1, z2)) -> cons(average(z1, z1), high(z0, z2))
   average(z0, z1) -> if(ge(z0, z1), z0, z1)
   if(true, z0, z1) -> averIter(z1, z0, z1)
   if(false, z0, z1) -> averIter(z0, z1, z0)
   averIter(z0, z1, z2) -> ifIter(ge(z0, z1), z0, z1, z2)
   ifIter(true, z0, z1, z2) -> z2
   ifIter(false, z0, z1, z2) -> averIter(plus(z0, s(s(s(0)))), plus(z1, s(0)), plus(z2, s(0)))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, 0) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(z0)) -> c1(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(z0, s(0)))))
   IFITER(false, 0, 0, x1) -> c1(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(z0), 0, x1) -> c1(AVERITER(s(plus(z0, s(s(s(0))))), s(0), plus(x1, s(0))))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c22(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c22(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c9(QUICKSORT(low(error, nil)))
   IFQUICK(false, cons(x0, cons(z1, z2))) -> c23(QUICKSORT(if_low(ge(z1, head(cons(x0, cons(z1, z2)))), head(cons(x0, cons(z1, z2))), cons(z1, z2))), LOW(head(cons(x0, cons(z1, z2))), tail(cons(x0, cons(z1, z2)))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(x0, nil)) -> c23(LOW(head(cons(x0, nil)), tail(cons(x0, nil))))
   IFQUICK(false, nil) -> c10(QUICKSORT(low(error, nil)))
S tuples:
   IF(true, z0, z1) -> c2(AVERITER(z1, z0, z1))
   IF(false, z0, z1) -> c3(AVERITER(z0, z1, z0))
   IF_LOW(false, z0, cons(z1, z2)) -> c13(LOW(z0, z2))
   IF_LOW(true, z0, cons(z1, z2)) -> c14(LOW(z0, z2))
   IF_HIGH(false, z0, cons(z1, z2)) -> c17(HIGH(z0, z2))
   IF_HIGH(true, z0, cons(z1, z2)) -> c18(AVERAGE(z1, z1))
   IF_HIGH(true, z0, cons(z1, z2)) -> c19(HIGH(z0, z2))
   PLUS(s(z0), z1) -> c28(PLUS(z0, z1))
   GE(s(z0), s(z1)) -> c37(GE(z0, z1))
   AVERAGE(z0, z1) -> c5(GE(z0, z1))
   AVERITER(0, s(z0), x2) -> c4(IFITER(false, 0, s(z0), x2))
   LOW(0, cons(z0, x2)) -> c12(IF_LOW(true, 0, cons(z0, x2)))
   LOW(s(z0), cons(0, x2)) -> c12(IF_LOW(false, s(z0), cons(0, x2)))
   HIGH(0, cons(z0, x2)) -> c16(IF_HIGH(true, 0, cons(z0, x2)))
   HIGH(s(z0), cons(0, x2)) -> c16(IF_HIGH(false, s(z0), cons(0, x2)))
   QUICKSORT(cons(z0, z1)) -> c20(IFQUICK(false, cons(z0, z1)))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(z0, tail(cons(z0, z1)))), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   AVERAGE(z0, 0) -> c5(IF(true, z0, 0))
   AVERAGE(s(z0), s(z1)) -> c5(IF(ge(z0, z1), s(z0), s(z1)))
   AVERITER(s(0), s(s(z0)), x2) -> c4(IFITER(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0))))
   AVERITER(s(s(z0)), s(s(z1)), x2) -> c4(IFITER(ge(z0, z1), s(s(z0)), s(s(z1)), x2), GE(s(s(z0)), s(s(z1))))
   AVERITER(s(x0), s(x1), x2) -> c4(GE(s(x0), s(x1)))
   AVERITER(s(z0), s(0), x2) -> c4(GE(s(z0), s(0)))
   IFITER(false, x0, s(z0), 0) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, 0) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, 0) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)))
   IFITER(false, x0, x1, s(0)) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, x1, s(s(z0))) -> c6(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(z0), s(x2)) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), x1, s(x2)) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, x1, s(x2)) -> c6(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))))
   IFITER(false, x0, s(0), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, x0, s(s(z0)), x2) -> c6(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x0, s(s(s(0)))))
   IFITER(false, s(z0), s(x1), x2) -> c6(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(z0), s(s(s(0)))))
   IFITER(false, 0, s(x1), x2) -> c6(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))))
   IFITER(false, s(0), x1, x2) -> c6(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(0), s(s(s(0)))))
   IFITER(false, s(s(z0)), x1, x2) -> c6(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(s(s(z0)), s(s(s(0)))))
   IFITER(false, s(x0), x1, x2) -> c6(PLUS(s(x0), s(s(s(0)))))
   IFITER(false, x0, s(z0), 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(0)), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, 0) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, 0) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(0)), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c7(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(x1, s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(z0), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c7(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(x1, s(0)))
   IFITER(false, x0, s(0), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(s(s(z0)), s(0)))
   IFITER(false, 0, s(x1), x2) -> c7(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c7(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(s(x1), s(0)))
   IFITER(false, x0, s(x1), x2) -> c7(PLUS(s(x1), s(0)))
   IFITER(false, s(0), x1, x2) -> c7(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c7(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x1, s(0)))
   IFITER(false, x0, x1, s(0)) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(0))), PLUS(s(0), s(0)))
   IFITER(false, x0, x1, s(s(z0))) -> c8(AVERITER(plus(x0, s(s(s(0)))), plus(x1, s(0)), s(s(plus(z0, s(0))))), PLUS(s(s(z0)), s(0)))
   IFITER(false, x0, s(z0), s(x2)) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(z0, s(0))), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, 0, x1, s(x2)) -> c8(AVERITER(s(s(s(0))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, s(z0), x1, s(x2)) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), plus(x1, s(0)), s(plus(x2, s(0)))), PLUS(s(x2), s(0)))
   IFITER(false, x0, x1, s(x2)) -> c8(PLUS(s(x2), s(0)))
   IFITER(false, x0, s(0), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(s(z0)), x2) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(s(plus(z0, s(0)))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, 0, s(x1), x2) -> c8(AVERITER(s(s(s(0))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(z0), s(x1), x2) -> c8(AVERITER(s(plus(z0, s(s(s(0))))), s(plus(x1, s(0))), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, x0, s(x1), 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(plus(x1, s(0))), s(0)))
   IFITER(false, 0, x0, 0) -> c8(AVERITER(s(s(s(0))), plus(x0, s(0)), s(0)))
   IFITER(false, s(0), x1, x2) -> c8(AVERITER(s(s(s(s(0)))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(s(z0)), x1, x2) -> c8(AVERITER(s(s(plus(z0, s(s(s(0)))))), plus(x1, s(0)), plus(x2, s(0))), PLUS(x2, s(0)))
   IFITER(false, s(x0), x1, 0) -> c8(AVERITER(s(plus(x0, s(s(s(0))))), plus(x1, s(0)), s(0)))
   LOW(s(0), cons(s(z0), x2)) -> c12(IF_LOW(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   LOW(s(s(z0)), cons(s(0), x2)) -> c12(IF_LOW(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   LOW(s(s(z1)), cons(s(s(z0)), x2)) -> c12(IF_LOW(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(0), cons(s(z0), x2)) -> c16(IF_HIGH(true, s(0), cons(s(z0), x2)), GE(s(z0), s(0)))
   HIGH(s(s(z0)), cons(s(0), x2)) -> c16(IF_HIGH(false, s(s(z0)), cons(s(0), x2)), GE(s(0), s(s(z0))))
   HIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c16(IF_HIGH(ge(z0, z1), s(s(z1)), cons(s(s(z0)), x2)), GE(s(s(z0)), s(s(z1))))
   HIGH(s(x0), cons(s(x1), x2)) -> c16(GE(s(x1), s(x0)))
K tuples:
   IFQUICK(false, nil) -> c11(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c11(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c11(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c25(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(head(nil), nil)))
   IFQUICK(false, nil) -> c15(HIGH(head(nil), tail(nil)))
   IFQUICK(false, nil) -> c15(QUICKSORT(high(error, tail(nil))))
   IFQUICK(false, cons(z0, z1)) -> c26(QUICKSORT(high(head(cons(z0, z1)), z1)), HIGH(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFITER(false, x0, 0, 0) -> c21(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c24(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, s(x0), 0, x2) -> c27(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, 0, 0, x1) -> c6(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, x0, 0, 0) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFITER(false, x0, 0, s(x2)) -> c7(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c7(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c7(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, s(x2)) -> c29(AVERITER(plus(x0, s(s(s(0)))), s(0), s(plus(x2, s(0)))))
   IFITER(false, 0, 0, x1) -> c30(AVERITER(s(s(s(0))), s(0), plus(x1, s(0))))
   IFITER(false, s(x0), 0, x2) -> c31(AVERITER(s(plus(x0, s(s(s(0))))), s(0), plus(x2, s(0))))
   IFITER(false, x0, 0, 0) -> c8(AVERITER(plus(x0, s(s(s(0)))), s(0), s(0)))
   IFQUICK(false, cons(z0, z1)) -> c22(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   IFQUICK(false, cons(z0, z1)) -> c23(QUICKSORT(low(z0, z1)), LOW(head(cons(z0, z1)), tail(cons(z0, z1))))
   LOW(s(x0), cons(s(x1), x2)) -> c12(GE(s(x1), s(x0)))
Defined Rule Symbols:   low_2, head_1, tail_1, if_low_3, ge_2, high_2, if_high_3, average_2, if_3, averIter_3, ifIter_4, plus_2

Defined Pair Symbols:   IF_3, IF_LOW_3, IF_HIGH_3, PLUS_2, GE_2, AVERAGE_2, AVERITER_3, LOW_2, HIGH_2, QUICKSORT_1, IFQUICK_2, IFITER_4

Compound Symbols:   c2_1, c3_1, c13_1, c14_1, c17_1, c18_1, c19_1, c28_1, c37_1, c5_1, c4_1, c12_1, c16_1, c20_1, c25_2, c11_1, c26_2, c15_1, c4_2, c6_2, c6_1, c21_1, c24_1, c27_1, c_1, c7_2, c7_1, c8_2, c8_1, c29_1, c30_1, c31_1, c1_1, c12_2, c16_2, c22_2, c22_1, c9_1, c23_2, c23_1, c10_1