KILLED
proof of input_xuXsq9puc3.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 11 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 333 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 21 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 82 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 7 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 7 ms]
            (34) typed CpxTrs
(35) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(36) CpxWeightedTrs
    (37) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxWeightedTrs
    (39) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxTypedWeightedTrs
    (41) CompletionProof [UPPER BOUND(ID), 0 ms]
    (42) CpxTypedWeightedCompleteTrs
        (43) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
        (46) CpxRNTS
        (47) InliningProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CpxRNTS
        (51) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CpxRNTS
        (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 185 ms]
        (56) CpxRNTS
        (57) IntTrsBoundProof [UPPER BOUND(ID), 48 ms]
        (58) CpxRNTS
        (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 486 ms]
        (62) CpxRNTS
        (63) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
        (64) CpxRNTS
        (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
        (67) IntTrsBoundProof [UPPER BOUND(ID), 221 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
        (70) CpxRNTS
        (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (72) CpxRNTS
        (73) IntTrsBoundProof [UPPER BOUND(ID), 341 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
        (76) CpxRNTS
        (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (78) CpxRNTS
        (79) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
        (80) CpxRNTS
        (81) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
        (82) CpxRNTS
        (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (84) CpxRNTS
        (85) IntTrsBoundProof [UPPER BOUND(ID), 1037 ms]
        (86) CpxRNTS
        (87) IntTrsBoundProof [UPPER BOUND(ID), 638 ms]
        (88) CpxRNTS
    (89) CompletionProof [UPPER BOUND(ID), 0 ms]
    (90) CpxTypedWeightedCompleteTrs
        (91) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 8 ms]
        (92) CpxRNTS
(93) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(94) CdtProblem
    (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CpxRelTRS
        (103) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (104) CpxTRS
        (105) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CpxWeightedTrs
            (107) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (108) CpxWeightedTrs
            (109) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (110) CpxTypedWeightedTrs
            (111) CompletionProof [UPPER BOUND(ID), 0 ms]
            (112) CpxTypedWeightedCompleteTrs
                (113) NarrowingProof [BOTH BOUNDS(ID, ID), 24 ms]
                (114) CpxTypedWeightedCompleteTrs
                (115) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (116) CpxRNTS
                (117) InliningProof [UPPER BOUND(ID), 0 ms]
                (118) CpxRNTS
                (119) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (120) CpxRNTS
                (121) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (122) CpxRNTS
                (123) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (124) CpxRNTS
                (125) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
                (126) CpxRNTS
                (127) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
                (128) CpxRNTS
                (129) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (130) CpxRNTS
                (131) IntTrsBoundProof [UPPER BOUND(ID), 189 ms]
                (132) CpxRNTS
                (133) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
                (134) CpxRNTS
                (135) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (136) CpxRNTS
                (137) IntTrsBoundProof [UPPER BOUND(ID), 391 ms]
                (138) CpxRNTS
                (139) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
                (140) CpxRNTS
                (141) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (142) CpxRNTS
                (143) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
                (144) CpxRNTS
                (145) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
                (146) CpxRNTS
                (147) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (148) CpxRNTS
                (149) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
                (150) CpxRNTS
                (151) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
                (152) CpxRNTS
                (153) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (154) CpxRNTS
                (155) IntTrsBoundProof [UPPER BOUND(ID), 342 ms]
                (156) CpxRNTS
                (157) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
                (158) CpxRNTS
                (159) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (160) CpxRNTS
                (161) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
                (162) CpxRNTS
                (163) IntTrsBoundProof [UPPER BOUND(ID), 145 ms]
                (164) CpxRNTS
                (165) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (166) CpxRNTS
                (167) IntTrsBoundProof [UPPER BOUND(ID), 28 ms]
                (168) CpxRNTS
                (169) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
                (170) CpxRNTS
                (171) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (172) CpxRNTS
                (173) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
                (174) CpxRNTS
                (175) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
                (176) CpxRNTS
                (177) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (178) CpxRNTS
                (179) IntTrsBoundProof [UPPER BOUND(ID), 1511 ms]
                (180) CpxRNTS
                (181) IntTrsBoundProof [UPPER BOUND(ID), 821 ms]
                (182) CpxRNTS
            (183) CompletionProof [UPPER BOUND(ID), 0 ms]
            (184) CpxTypedWeightedCompleteTrs
                (185) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (186) CpxRNTS
    (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
        (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms]
        (190) CdtProblem
        (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms]
        (192) CdtProblem
        (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (194) CdtProblem
        (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (196) CdtProblem
        (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms]
        (198) CdtProblem
        (199) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (200) CdtProblem
        (201) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (202) CdtProblem
        (203) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 117 ms]
        (204) CdtProblem
        (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (206) CdtProblem
        (207) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (208) CdtProblem
        (209) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (210) CdtProblem
        (211) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 126 ms]
        (212) CdtProblem
        (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (214) CdtProblem
        (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (216) CdtProblem
        (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (218) CdtProblem
        (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (220) CdtProblem
        (221) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (222) CdtProblem
        (223) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (224) CdtProblem
        (225) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 200 ms]
        (226) CdtProblem
        (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (228) CdtProblem
        (229) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (230) CdtProblem
        (231) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (232) CdtProblem
        (233) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 192 ms]
        (234) CdtProblem
        (235) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (236) CdtProblem
        (237) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (238) CdtProblem
        (239) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (240) CdtProblem
        (241) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (242) CdtProblem
        (243) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 230 ms]
        (244) CdtProblem
        (245) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (246) CdtProblem
        (247) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (248) CdtProblem
        (249) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (250) CdtProblem
        (251) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 2 ms]
        (252) CdtProblem
        (253) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (254) CdtProblem
        (255) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (256) CdtProblem
        (257) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 207 ms]
        (258) CdtProblem
        (259) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms]
        (260) CdtProblem
        (261) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (262) CdtProblem
        (263) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 254 ms]
        (264) CdtProblem
        (265) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (266) CdtProblem
        (267) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (268) CdtProblem
        (269) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 35 ms]
        (270) CdtProblem
        (271) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (272) CdtProblem
        (273) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (274) CdtProblem
        (275) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (276) CdtProblem
        (277) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (278) CdtProblem
        (279) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (280) CdtProblem
        (281) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 273 ms]
        (282) CdtProblem
        (283) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (284) CdtProblem
        (285) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (286) CdtProblem
        (287) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (288) CdtProblem
        (289) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 300 ms]
        (290) CdtProblem
        (291) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (292) CdtProblem
        (293) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (294) CdtProblem
        (295) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (296) CdtProblem
        (297) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (298) CdtProblem
        (299) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (300) CdtProblem
        (301) CdtRewritingProof [BOTH BOUNDS(ID, ID), 15 ms]
        (302) CdtProblem
        (303) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (304) CdtProblem
        (305) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 12 ms]
        (306) CdtProblem
        (307) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (308) CdtProblem
        (309) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 16 ms]
        (310) CdtProblem
        (311) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms]
        (312) CdtProblem
        (313) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (314) CdtProblem
        (315) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 254 ms]
        (316) CdtProblem
        (317) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (318) CdtProblem
        (319) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (320) CdtProblem
        (321) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 198 ms]
        (322) CdtProblem
        (323) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 27 ms]
        (324) CdtProblem
        (325) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (326) CdtProblem
        (327) CdtRewritingProof [BOTH BOUNDS(ID, ID), 45 ms]
        (328) CdtProblem
        (329) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (330) CdtProblem
        (331) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (332) CdtProblem
        (333) CdtRewritingProof [BOTH BOUNDS(ID, ID), 40 ms]
        (334) CdtProblem
        (335) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (336) CdtProblem
        (337) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (338) CdtProblem
        (339) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (340) CdtProblem
        (341) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (342) CdtProblem
        (343) CdtRewritingProof [BOTH BOUNDS(ID, ID), 14 ms]
        (344) CdtProblem
        (345) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (346) CdtProblem
        (347) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (348) CdtProblem
        (349) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (350) CdtProblem
        (351) CdtRewritingProof [BOTH BOUNDS(ID, ID), 34 ms]
        (352) CdtProblem
        (353) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (354) 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:

   min(0, y) -> 0
   min(x, 0) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(0, y) -> y
   max(x, 0) -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

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:

   min(0', y) -> 0'
   min(x, 0') -> 0'
   min(s(x), s(y)) -> s(min(x, y))
   max(0', y) -> y
   max(x, 0') -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0') -> 0'
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

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:

   min(0, y) -> 0
   min(x, 0) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(0, y) -> y
   max(x, 0) -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

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:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, P_1, F_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10, c11_3, c12_3, c13_4


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

(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:

   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))

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

   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

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:

   MIN(0', z0) -> c
   MIN(z0, 0') -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0', z0) -> c3
   MAX(z0, 0') -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0') -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0') -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))

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

   min(0', z0) -> 0'
   min(z0, 0') -> 0'
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0', z0) -> z0
   max(z0, 0') -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0') -> 0'
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0') -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MIN, MAX, TWICE, -', F, -, max, min, twice, f

They will be analysed ascendingly in the following order:
MIN < F
MAX < F
TWICE < F
-' < F
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

(14)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
MIN, MAX, TWICE, -', F, -, max, min, twice, f

They will be analysed ascendingly in the following order:
MIN < F
MAX < F
TWICE < F
-' < F
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)

Induction Base:
MIN(gen_0':s10_14(0), gen_0':s10_14(0)) ->_R^Omega(1)
c

Induction Step:
MIN(gen_0':s10_14(+(n16_14, 1)), gen_0':s10_14(+(n16_14, 1))) ->_R^Omega(1)
c2(MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14))) ->_IH
c2(gen_c:c1:c29_14(c17_14))

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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
MIN, MAX, TWICE, -', F, -, max, min, twice, f

They will be analysed ascendingly in the following order:
MIN < F
MAX < F
TWICE < F
-' < F
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
MAX, TWICE, -', F, -, max, min, twice, f

They will be analysed ascendingly in the following order:
MAX < F
TWICE < F
-' < F
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)

Induction Base:
MAX(gen_0':s10_14(0), gen_0':s10_14(0)) ->_R^Omega(1)
c3

Induction Step:
MAX(gen_0':s10_14(+(n536_14, 1)), gen_0':s10_14(+(n536_14, 1))) ->_R^Omega(1)
c5(MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14))) ->_IH
c5(gen_c3:c4:c511_14(c537_14))

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

(22)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
TWICE, -', F, -, max, min, twice, f

They will be analysed ascendingly in the following order:
TWICE < F
-' < F
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)

Induction Base:
TWICE(gen_0':s10_14(0)) ->_R^Omega(1)
c6

Induction Step:
TWICE(gen_0':s10_14(+(n1131_14, 1))) ->_R^Omega(1)
c7(TWICE(gen_0':s10_14(n1131_14))) ->_IH
c7(gen_c6:c712_14(c1132_14))

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

(24)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
-', F, -, max, min, twice, f

They will be analysed ascendingly in the following order:
-' < F
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14)) -> gen_c8:c913_14(n1566_14), rt in Omega(1 + n1566_14)

Induction Base:
-'(gen_0':s10_14(0), gen_0':s10_14(0)) ->_R^Omega(1)
c8

Induction Step:
-'(gen_0':s10_14(+(n1566_14, 1)), gen_0':s10_14(+(n1566_14, 1))) ->_R^Omega(1)
c9(-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14))) ->_IH
c9(gen_c8:c913_14(c1567_14))

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:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)
-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14)) -> gen_c8:c913_14(n1566_14), rt in Omega(1 + n1566_14)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
-, F, max, min, twice, f

They will be analysed ascendingly in the following order:
- < F
max < F
min < F
twice < F
- < f
max < f
min < f
twice < f

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s10_14(n2212_14), gen_0':s10_14(n2212_14)) -> gen_0':s10_14(0), rt in Omega(0)

Induction Base:
-(gen_0':s10_14(0), gen_0':s10_14(0)) ->_R^Omega(0)
gen_0':s10_14(0)

Induction Step:
-(gen_0':s10_14(+(n2212_14, 1)), gen_0':s10_14(+(n2212_14, 1))) ->_R^Omega(0)
-(gen_0':s10_14(n2212_14), gen_0':s10_14(n2212_14)) ->_IH
gen_0':s10_14(0)

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

(28)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)
-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14)) -> gen_c8:c913_14(n1566_14), rt in Omega(1 + n1566_14)
-(gen_0':s10_14(n2212_14), gen_0':s10_14(n2212_14)) -> gen_0':s10_14(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
max, F, min, twice, f

They will be analysed ascendingly in the following order:
max < F
min < F
twice < F
max < f
min < f
twice < f

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
max(gen_0':s10_14(n3084_14), gen_0':s10_14(n3084_14)) -> gen_0':s10_14(n3084_14), rt in Omega(0)

Induction Base:
max(gen_0':s10_14(0), gen_0':s10_14(0)) ->_R^Omega(0)
gen_0':s10_14(0)

Induction Step:
max(gen_0':s10_14(+(n3084_14, 1)), gen_0':s10_14(+(n3084_14, 1))) ->_R^Omega(0)
s(max(gen_0':s10_14(n3084_14), gen_0':s10_14(n3084_14))) ->_IH
s(gen_0':s10_14(c3085_14))

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

(30)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)
-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14)) -> gen_c8:c913_14(n1566_14), rt in Omega(1 + n1566_14)
-(gen_0':s10_14(n2212_14), gen_0':s10_14(n2212_14)) -> gen_0':s10_14(0), rt in Omega(0)
max(gen_0':s10_14(n3084_14), gen_0':s10_14(n3084_14)) -> gen_0':s10_14(n3084_14), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
min, F, twice, f

They will be analysed ascendingly in the following order:
min < F
twice < F
min < f
twice < f

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_0':s10_14(n4006_14), gen_0':s10_14(n4006_14)) -> gen_0':s10_14(n4006_14), rt in Omega(0)

Induction Base:
min(gen_0':s10_14(0), gen_0':s10_14(0)) ->_R^Omega(0)
0'

Induction Step:
min(gen_0':s10_14(+(n4006_14, 1)), gen_0':s10_14(+(n4006_14, 1))) ->_R^Omega(0)
s(min(gen_0':s10_14(n4006_14), gen_0':s10_14(n4006_14))) ->_IH
s(gen_0':s10_14(c4007_14))

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

(32)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)
-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14)) -> gen_c8:c913_14(n1566_14), rt in Omega(1 + n1566_14)
-(gen_0':s10_14(n2212_14), gen_0':s10_14(n2212_14)) -> gen_0':s10_14(0), rt in Omega(0)
max(gen_0':s10_14(n3084_14), gen_0':s10_14(n3084_14)) -> gen_0':s10_14(n3084_14), rt in Omega(0)
min(gen_0':s10_14(n4006_14), gen_0':s10_14(n4006_14)) -> gen_0':s10_14(n4006_14), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
twice, F, f

They will be analysed ascendingly in the following order:
twice < F
twice < f

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
twice(gen_0':s10_14(n4610_14)) -> gen_0':s10_14(*(2, n4610_14)), rt in Omega(0)

Induction Base:
twice(gen_0':s10_14(0)) ->_R^Omega(0)
0'

Induction Step:
twice(gen_0':s10_14(+(n4610_14, 1))) ->_R^Omega(0)
s(s(twice(gen_0':s10_14(n4610_14)))) ->_IH
s(s(gen_0':s10_14(*(2, c4611_14))))

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

(34)
Obligation:
Innermost TRS:
Rules:
MIN(0', z0) -> c
MIN(z0, 0') -> c1
MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
MAX(0', z0) -> c3
MAX(z0, 0') -> c4
MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
TWICE(0') -> c6
TWICE(s(z0)) -> c7(TWICE(z0))
-'(z0, 0') -> c8
-'(s(z0), s(z1)) -> c9(-'(z0, z1))
P(s(z0)) -> c10
F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
min(0', z0) -> 0'
min(z0, 0') -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
max(0', z0) -> z0
max(z0, 0') -> z0
max(s(z0), s(z1)) -> s(max(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
p(s(z0)) -> z0
f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

Types:
MIN :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
MAX :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
TWICE :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
-' :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
P :: 0':s -> c10
c10 :: c10
F :: 0':s -> 0':s -> c11:c12:c13
c11 :: c11:c12:c13 -> c8:c9 -> c3:c4:c5 -> c11:c12:c13
- :: 0':s -> 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
twice :: 0':s -> 0':s
c12 :: c11:c12:c13 -> c8:c9 -> c:c1:c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c10 -> c6:c7 -> c:c1:c2 -> c11:c12:c13
f :: 0':s -> 0':s -> f
hole_c:c1:c21_14 :: c:c1:c2
hole_0':s2_14 :: 0':s
hole_c3:c4:c53_14 :: c3:c4:c5
hole_c6:c74_14 :: c6:c7
hole_c8:c95_14 :: c8:c9
hole_c106_14 :: c10
hole_c11:c12:c137_14 :: c11:c12:c13
hole_f8_14 :: f
gen_c:c1:c29_14 :: Nat -> c:c1:c2
gen_0':s10_14 :: Nat -> 0':s
gen_c3:c4:c511_14 :: Nat -> c3:c4:c5
gen_c6:c712_14 :: Nat -> c6:c7
gen_c8:c913_14 :: Nat -> c8:c9
gen_c11:c12:c1314_14 :: Nat -> c11:c12:c13


Lemmas:
MIN(gen_0':s10_14(n16_14), gen_0':s10_14(n16_14)) -> gen_c:c1:c29_14(n16_14), rt in Omega(1 + n16_14)
MAX(gen_0':s10_14(n536_14), gen_0':s10_14(n536_14)) -> gen_c3:c4:c511_14(n536_14), rt in Omega(1 + n536_14)
TWICE(gen_0':s10_14(n1131_14)) -> gen_c6:c712_14(n1131_14), rt in Omega(1 + n1131_14)
-'(gen_0':s10_14(n1566_14), gen_0':s10_14(n1566_14)) -> gen_c8:c913_14(n1566_14), rt in Omega(1 + n1566_14)
-(gen_0':s10_14(n2212_14), gen_0':s10_14(n2212_14)) -> gen_0':s10_14(0), rt in Omega(0)
max(gen_0':s10_14(n3084_14), gen_0':s10_14(n3084_14)) -> gen_0':s10_14(n3084_14), rt in Omega(0)
min(gen_0':s10_14(n4006_14), gen_0':s10_14(n4006_14)) -> gen_0':s10_14(n4006_14), rt in Omega(0)
twice(gen_0':s10_14(n4610_14)) -> gen_0':s10_14(*(2, n4610_14)), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_14(0) <=> c
gen_c:c1:c29_14(+(x, 1)) <=> c2(gen_c:c1:c29_14(x))
gen_0':s10_14(0) <=> 0'
gen_0':s10_14(+(x, 1)) <=> s(gen_0':s10_14(x))
gen_c3:c4:c511_14(0) <=> c3
gen_c3:c4:c511_14(+(x, 1)) <=> c5(gen_c3:c4:c511_14(x))
gen_c6:c712_14(0) <=> c6
gen_c6:c712_14(+(x, 1)) <=> c7(gen_c6:c712_14(x))
gen_c8:c913_14(0) <=> c8
gen_c8:c913_14(+(x, 1)) <=> c9(gen_c8:c913_14(x))
gen_c11:c12:c1314_14(0) <=> hole_c11:c12:c137_14
gen_c11:c12:c1314_14(+(x, 1)) <=> c11(gen_c11:c12:c1314_14(x), c8, c3)


The following defined symbols remain to be analysed:
F, f
----------------------------------------

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

(36)
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:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   -(x, 0) -> x [1]
   -(s(x), s(y)) -> -(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]

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

(37) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

(38)
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:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]

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

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

(40)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]

The TRS has the following type information:
min :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
max :: 0:s -> 0:s -> 0:s
twice :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> f

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

(41) 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:

   f_2

(c) The following functions are completely defined:

   minus_2
   max_2
   min_2
   p_1
   twice_1

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]

And the following fresh constants:    const

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

(42)
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:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]
   minus(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
min :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
max :: 0:s -> 0:s -> 0:s
twice :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> f
const :: f

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

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

(44)
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:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(0), s(y)) -> f(minus(s(max(0, y)), s(min(0, y))), p(twice(0))) [4]
   f(s(x), s(0)) -> f(minus(s(max(x, 0)), s(min(x, 0))), p(twice(0))) [4]
   f(s(s(x')), s(s(y'))) -> f(minus(s(max(s(x'), s(y'))), s(min(s(x'), s(y')))), p(twice(s(min(x', y'))))) [4]
   minus(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
min :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
max :: 0:s -> 0:s -> 0:s
twice :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> f
const :: f

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

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

   0 => 0
   const => 0

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(x, 0), 1 + min(x, 0)), p(twice(0))) :|: x >= 0, z' = 1 + 0, z = 1 + x
   f(z, z') -{ 4 }-> f(minus(1 + max(0, y), 1 + min(0, y)), p(twice(0))) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + x', 1 + y'), 1 + min(1 + x', 1 + y')), p(twice(1 + min(x', y')))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x


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

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

   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(x, 0), 1 + min(x, 0)), p(twice(0))) :|: x >= 0, z' = 1 + 0, z = 1 + x
   f(z, z') -{ 4 }-> f(minus(1 + max(0, y), 1 + min(0, y)), p(twice(0))) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + x', 1 + y'), 1 + min(1 + x', 1 + y')), p(twice(1 + min(x', y')))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x


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

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0


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

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

   { twice }
   { max }
   { minus }
   { min }
   { p }
   { f }

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f}

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

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f}

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: ?, size: O(n^1) [2*z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

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

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

   f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: ?, size: O(n^1) [z + z']

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']

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

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

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

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: ?, size: O(n^1) [z]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

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

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

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: ?, size: O(n^1) [z]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]

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

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

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

   f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]

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

(79) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: ?, size: O(n^1) [z]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: p
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:

   f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(85) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: ?, size: O(1) [0]

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

(87) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: INF, size: O(1) [0]

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

(89) 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:

   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]
   min(v0, v1) -> null_min [0]
   max(v0, v1) -> null_max [0]
   twice(v0) -> null_twice [0]

And the following fresh constants:    null_minus, null_p, null_f, null_min, null_max, null_twice

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

(90)
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:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]
   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]
   min(v0, v1) -> null_min [0]
   max(v0, v1) -> null_max [0]
   twice(v0) -> null_twice [0]

The TRS has the following type information:
min :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
0 :: 0:s:null_minus:null_p:null_min:null_max:null_twice
s :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
max :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
twice :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
minus :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
p :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
f :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> null_f
null_minus :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_p :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_f :: null_f
null_min :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_max :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_twice :: 0:s:null_minus:null_p:null_min:null_max:null_twice

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

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

   0 => 0
   null_minus => 0
   null_p => 0
   null_f => 0
   null_min => 0
   null_max => 0
   null_twice => 0

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> f(minus(max(1 + x, 1 + y), min(1 + x, 1 + y)), p(twice(min(x, y)))) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

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

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, P_1, F_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10, c11_3, c12_3, c13_4


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

(95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   P(s(z0)) -> c10
   TWICE(0) -> c6
   -'(z0, 0) -> c8
   MAX(z0, 0) -> c4
   MIN(z0, 0) -> c1
   MAX(0, z0) -> c3
   MIN(0, z0) -> c

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_4


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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3


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

(99) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3


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

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

(102)
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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

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

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))

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

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

(104)
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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))

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

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

(106)
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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1]
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1]
   TWICE(s(z0)) -> c7(TWICE(z0)) [1]
   -'(s(z0), s(z1)) -> c9(-'(z0, z1)) [1]
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   -(z0, 0) -> z0 [0]
   -(s(z0), s(z1)) -> -(z0, z1) [0]
   max(s(z0), s(z1)) -> s(max(z0, z1)) [0]
   max(0, z0) -> z0 [0]
   max(z0, 0) -> z0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]

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

(107) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

(108)
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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1]
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1]
   TWICE(s(z0)) -> c7(TWICE(z0)) [1]
   -'(s(z0), s(z1)) -> c9(-'(z0, z1)) [1]
   F(s(z0), s(z1)) -> c11(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c12(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c13(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   max(s(z0), s(z1)) -> s(max(z0, z1)) [0]
   max(0, z0) -> z0 [0]
   max(z0, 0) -> z0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]

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

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

(110)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1]
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1]
   TWICE(s(z0)) -> c7(TWICE(z0)) [1]
   -'(s(z0), s(z1)) -> c9(-'(z0, z1)) [1]
   F(s(z0), s(z1)) -> c11(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c12(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c13(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   max(s(z0), s(z1)) -> s(max(z0, z1)) [0]
   max(0, z0) -> z0 [0]
   max(z0, 0) -> z0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]

The TRS has the following type information:
MIN :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MAX :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
TWICE :: s:0 -> c7
c7 :: c7 -> c7
-' :: s:0 -> s:0 -> c9
c9 :: c9 -> c9
F :: s:0 -> s:0 -> c11:c12:c13
c11 :: c11:c12:c13 -> c9 -> c5 -> c11:c12:c13
minus :: s:0 -> s:0 -> s:0
max :: s:0 -> s:0 -> s:0
min :: s:0 -> s:0 -> s:0
p :: s:0 -> s:0
twice :: s:0 -> s:0
c12 :: c11:c12:c13 -> c9 -> c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c7 -> c2 -> c11:c12:c13
0 :: s:0

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

(111) 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:

   MIN_2
   MAX_2
   TWICE_1
   -'_2
   F_2

(c) The following functions are completely defined:

   minus_2
   max_2
   min_2
   p_1
   twice_1

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]
   max(v0, v1) -> 0 [0]
   min(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]
   twice(v0) -> 0 [0]

And the following fresh constants:    const, const1, const2, const3, const4

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

(112)
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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1]
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1]
   TWICE(s(z0)) -> c7(TWICE(z0)) [1]
   -'(s(z0), s(z1)) -> c9(-'(z0, z1)) [1]
   F(s(z0), s(z1)) -> c11(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c12(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c13(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   max(s(z0), s(z1)) -> s(max(z0, z1)) [0]
   max(0, z0) -> z0 [0]
   max(z0, 0) -> z0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]
   minus(v0, v1) -> 0 [0]
   max(v0, v1) -> 0 [0]
   min(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]
   twice(v0) -> 0 [0]

The TRS has the following type information:
MIN :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MAX :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
TWICE :: s:0 -> c7
c7 :: c7 -> c7
-' :: s:0 -> s:0 -> c9
c9 :: c9 -> c9
F :: s:0 -> s:0 -> c11:c12:c13
c11 :: c11:c12:c13 -> c9 -> c5 -> c11:c12:c13
minus :: s:0 -> s:0 -> s:0
max :: s:0 -> s:0 -> s:0
min :: s:0 -> s:0 -> s:0
p :: s:0 -> s:0
twice :: s:0 -> s:0
c12 :: c11:c12:c13 -> c9 -> c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c7 -> c2 -> c11:c12:c13
0 :: s:0
const :: c2
const1 :: c5
const2 :: c7
const3 :: c9
const4 :: c11:c12:c13

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

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

(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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1]
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1]
   TWICE(s(z0)) -> c7(TWICE(z0)) [1]
   -'(s(z0), s(z1)) -> c9(-'(z0, z1)) [1]
   F(s(z0), s(z1)) -> c11(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c12(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c13(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   max(s(z0), s(z1)) -> s(max(z0, z1)) [0]
   max(0, z0) -> z0 [0]
   max(z0, 0) -> z0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]
   minus(v0, v1) -> 0 [0]
   max(v0, v1) -> 0 [0]
   min(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]
   twice(v0) -> 0 [0]

The TRS has the following type information:
MIN :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
MAX :: s:0 -> s:0 -> c5
c5 :: c5 -> c5
TWICE :: s:0 -> c7
c7 :: c7 -> c7
-' :: s:0 -> s:0 -> c9
c9 :: c9 -> c9
F :: s:0 -> s:0 -> c11:c12:c13
c11 :: c11:c12:c13 -> c9 -> c5 -> c11:c12:c13
minus :: s:0 -> s:0 -> s:0
max :: s:0 -> s:0 -> s:0
min :: s:0 -> s:0 -> s:0
p :: s:0 -> s:0
twice :: s:0 -> s:0
c12 :: c11:c12:c13 -> c9 -> c2 -> c11:c12:c13
c13 :: c11:c12:c13 -> c7 -> c2 -> c11:c12:c13
0 :: s:0
const :: c2
const1 :: c5
const2 :: c7
const3 :: c9
const4 :: c11:c12:c13

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

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

   0 => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

(116)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + -'(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)) + MIN(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + -'(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)) + MAX(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MAX(z, z') -{ 1 }-> 1 + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0
   max(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   max(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 0 }-> 1 + max(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0


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

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

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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + -'(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)) + MIN(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + -'(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)) + MAX(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MAX(z, z') -{ 1 }-> 1 + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0
   max(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   max(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 0 }-> 1 + max(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0


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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0


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

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

   { twice }
   { -' }
   { minus }
   { TWICE }
   { MIN }
   { max }
   { min }
   { p }
   { MAX }
   { F }

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {-'}, {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}

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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {-'}, {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}

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

(125) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {-'}, {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: ?, size: O(n^1) [2*z]

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

(127) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: twice
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:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {-'}, {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]

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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {-'}, {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]

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

(131) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {-'}, {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: ?, size: O(1) [0]

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

(133) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: -'
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:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

(137) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: ?, size: O(n^1) [z]

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

(139) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

(143) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {TWICE}, {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: ?, size: O(1) [0]

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

(145) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]

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

(149) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {MIN}, {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: ?, size: O(1) [0]

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

(151) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MIN(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + z' }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]

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

(155) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + z' }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: ?, size: O(n^1) [z + z']

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

(157) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + z' }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(max(1 + (z - 1), 1 + (z' - 1)), min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + max(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(s10, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + s4 :|: s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(s6, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(s7, min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + z' }-> 1 + F(minus(s8, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(s9, min(1 + (z - 1), 1 + (z' - 1))) + s3 :|: s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']

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

(161) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(s10, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + s4 :|: s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(s6, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(s7, min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + z' }-> 1 + F(minus(s8, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(s9, min(1 + (z - 1), 1 + (z' - 1))) + s3 :|: s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: ?, size: O(n^1) [z']

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

(163) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: min
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:

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(s10, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + TWICE(min(z - 1, z' - 1)) + s4 :|: s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(s6, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(s7, min(1 + (z - 1), 1 + (z' - 1))) + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + z' }-> 1 + F(minus(s8, min(1 + (z - 1), 1 + (z' - 1))), p(twice(min(z - 1, z' - 1)))) + -'(s9, min(1 + (z - 1), 1 + (z' - 1))) + s3 :|: s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 }-> 1 + F(s13, p(s15)) + s17 + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, p(s21)) + s23 + s3 :|: s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, p(s27)) + s29 + s4 :|: s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']

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

(167) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 }-> 1 + F(s13, p(s15)) + s17 + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, p(s21)) + s23 + s3 :|: s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, p(s27)) + s29 + s4 :|: s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: ?, size: O(n^1) [z]

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

(169) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 }-> 1 + F(s13, p(s15)) + s17 + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, p(s21)) + s23 + s3 :|: s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, p(s27)) + s29 + s4 :|: s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 }-> 1 + F(s13, s30) + s17 + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s30 >= 0, s30 <= s15, s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, s31) + s23 + s3 :|: s31 >= 0, s31 <= s21, s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, s32) + s29 + s4 :|: s32 >= 0, s32 <= s27, s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]

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

(173) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(174)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 }-> 1 + F(s13, s30) + s17 + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s30 >= 0, s30 <= s15, s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, s31) + s23 + s3 :|: s31 >= 0, s31 <= s21, s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, s32) + s29 + s4 :|: s32 >= 0, s32 <= s27, s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {MAX}, {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MAX: runtime: ?, size: O(1) [0]

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

(175) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 }-> 1 + F(s13, s30) + s17 + MAX(1 + (z - 1), 1 + (z' - 1)) :|: s30 >= 0, s30 <= s15, s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, s31) + s23 + s3 :|: s31 >= 0, s31 <= s21, s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, s32) + s29 + s4 :|: s32 >= 0, s32 <= s27, s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ 1 }-> 1 + MAX(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MAX: runtime: O(n^1) [z'], size: O(1) [0]

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

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

(178)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 + z' }-> 1 + F(s13, s30) + s17 + s34 :|: s34 >= 0, s34 <= 0, s30 >= 0, s30 <= s15, s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, s31) + s23 + s3 :|: s31 >= 0, s31 <= s21, s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, s32) + s29 + s4 :|: s32 >= 0, s32 <= s27, s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ z' }-> 1 + s33 :|: s33 >= 0, s33 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MAX: runtime: O(n^1) [z'], size: O(1) [0]

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

(179) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(180)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 + z' }-> 1 + F(s13, s30) + s17 + s34 :|: s34 >= 0, s34 <= 0, s30 >= 0, s30 <= s15, s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, s31) + s23 + s3 :|: s31 >= 0, s31 <= s21, s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, s32) + s29 + s4 :|: s32 >= 0, s32 <= s27, s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ z' }-> 1 + s33 :|: s33 >= 0, s33 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MAX: runtime: O(n^1) [z'], size: O(1) [0]
  F: runtime: ?, size: O(1) [0]

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

(181) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(182)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s16 + z' }-> 1 + F(s13, s30) + s17 + s34 :|: s34 >= 0, s34 <= 0, s30 >= 0, s30 <= s15, s12 >= 0, s12 <= 1 + (z' - 1), s13 >= 0, s13 <= s6, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= 1 + (z' - 1), s17 >= 0, s17 <= 0, s6 >= 0, s6 <= 1 + (z' - 1) + (1 + (z - 1)), s7 >= 0, s7 <= 1 + (z' - 1) + (1 + (z - 1)), z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + z' }-> 1 + F(s19, s31) + s23 + s3 :|: s31 >= 0, s31 <= s21, s18 >= 0, s18 <= 1 + (z' - 1), s19 >= 0, s19 <= s8, s20 >= 0, s20 <= z' - 1, s21 >= 0, s21 <= 2 * s20, s22 >= 0, s22 <= 1 + (z' - 1), s23 >= 0, s23 <= 0, s8 >= 0, s8 <= 1 + (z' - 1) + (1 + (z - 1)), s9 >= 0, s9 <= 1 + (z' - 1) + (1 + (z - 1)), s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s28 + z' }-> 1 + F(s25, s32) + s29 + s4 :|: s32 >= 0, s32 <= s27, s24 >= 0, s24 <= 1 + (z' - 1), s25 >= 0, s25 <= s10, s26 >= 0, s26 <= z' - 1, s27 >= 0, s27 <= 2 * s26, s28 >= 0, s28 <= z' - 1, s29 >= 0, s29 <= 0, s10 >= 0, s10 <= 1 + (z' - 1) + (1 + (z - 1)), s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   MAX(z, z') -{ z' }-> 1 + s33 :|: s33 >= 0, s33 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
   max(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   max(z, z') -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s11 :|: s11 >= 0, s11 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  TWICE: runtime: O(n^1) [z], size: O(1) [0]
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  max: runtime: O(1) [0], size: O(n^1) [z + z']
  min: runtime: O(1) [0], size: O(n^1) [z']
  p: runtime: O(1) [0], size: O(n^1) [z]
  MAX: runtime: O(n^1) [z'], size: O(1) [0]
  F: runtime: INF, size: O(1) [0]

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

(183) 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:

   minus(v0, v1) -> null_minus [0]
   max(v0, v1) -> null_max [0]
   min(v0, v1) -> null_min [0]
   p(v0) -> null_p [0]
   twice(v0) -> null_twice [0]
   MIN(v0, v1) -> null_MIN [0]
   MAX(v0, v1) -> null_MAX [0]
   TWICE(v0) -> null_TWICE [0]
   -'(v0, v1) -> null_-' [0]
   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_minus, null_max, null_min, null_p, null_twice, null_MIN, null_MAX, null_TWICE, null_-', null_F

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

(184)
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:

   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) [1]
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) [1]
   TWICE(s(z0)) -> c7(TWICE(z0)) [1]
   -'(s(z0), s(z1)) -> c9(-'(z0, z1)) [1]
   F(s(z0), s(z1)) -> c11(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c12(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) [1]
   F(s(z0), s(z1)) -> c13(F(minus(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   max(s(z0), s(z1)) -> s(max(z0, z1)) [0]
   max(0, z0) -> z0 [0]
   max(z0, 0) -> z0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   min(0, z0) -> 0 [0]
   min(z0, 0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]
   minus(v0, v1) -> null_minus [0]
   max(v0, v1) -> null_max [0]
   min(v0, v1) -> null_min [0]
   p(v0) -> null_p [0]
   twice(v0) -> null_twice [0]
   MIN(v0, v1) -> null_MIN [0]
   MAX(v0, v1) -> null_MAX [0]
   TWICE(v0) -> null_TWICE [0]
   -'(v0, v1) -> null_-' [0]
   F(v0, v1) -> null_F [0]

The TRS has the following type information:
MIN :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> c2:null_MIN
s :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice
c2 :: c2:null_MIN -> c2:null_MIN
MAX :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> c5:null_MAX
c5 :: c5:null_MAX -> c5:null_MAX
TWICE :: s:0:null_minus:null_max:null_min:null_p:null_twice -> c7:null_TWICE
c7 :: c7:null_TWICE -> c7:null_TWICE
-' :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> c9:null_-'
c9 :: c9:null_-' -> c9:null_-'
F :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> c11:c12:c13:null_F
c11 :: c11:c12:c13:null_F -> c9:null_-' -> c5:null_MAX -> c11:c12:c13:null_F
minus :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice
max :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice
min :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice
p :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice
twice :: s:0:null_minus:null_max:null_min:null_p:null_twice -> s:0:null_minus:null_max:null_min:null_p:null_twice
c12 :: c11:c12:c13:null_F -> c9:null_-' -> c2:null_MIN -> c11:c12:c13:null_F
c13 :: c11:c12:c13:null_F -> c7:null_TWICE -> c2:null_MIN -> c11:c12:c13:null_F
0 :: s:0:null_minus:null_max:null_min:null_p:null_twice
null_minus :: s:0:null_minus:null_max:null_min:null_p:null_twice
null_max :: s:0:null_minus:null_max:null_min:null_p:null_twice
null_min :: s:0:null_minus:null_max:null_min:null_p:null_twice
null_p :: s:0:null_minus:null_max:null_min:null_p:null_twice
null_twice :: s:0:null_minus:null_max:null_min:null_p:null_twice
null_MIN :: c2:null_MIN
null_MAX :: c5:null_MAX
null_TWICE :: c7:null_TWICE
null_-' :: c9:null_-'
null_F :: c11:c12:c13:null_F

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

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

   0 => 0
   null_minus => 0
   null_max => 0
   null_min => 0
   null_p => 0
   null_twice => 0
   null_MIN => 0
   null_MAX => 0
   null_TWICE => 0
   null_-' => 0
   null_F => 0

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

(186)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + -'(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)) + MIN(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)), p(twice(min(z0, z1)))) + -'(max(1 + z0, 1 + z1), min(1 + z0, 1 + z1)) + MAX(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MAX(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MAX(z, z') -{ 1 }-> 1 + MAX(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   TWICE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0
   max(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   max(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 0 }-> 1 + max(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(187) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c11_3


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

(189) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c11_3, c12_3


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

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3


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

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2


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

(195) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) by 
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2


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

(197) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) by 
   F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2


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

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c11(-'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c11(-'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c11_2


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

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1


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

(203) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1


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

(205) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) by 
   F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1


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

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c11(-'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c11(-'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c11_2


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

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1


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

(211) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1]

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1


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

(213) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1


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

(215) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1


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

(217) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) by 
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1


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

(219) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) by 
   F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1


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

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c12(-'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c12(-'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c12_2


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

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1


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

(225) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1


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

(227) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) by 
   F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))

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

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1


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

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

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c12(-'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c12(-'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c12_2


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

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

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1


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

(233) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(234)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1


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

(235) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(236)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1


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

(237) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))

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

(238)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1


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

(239) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))

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

(240)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1


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

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

(242)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


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

(243) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(244)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


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

(245) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

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

(246)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


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

(247) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(248)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


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

(249) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))

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

(250)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1


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

(251) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(252)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1


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

(253) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) by 
   F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0)))
   F(s(0), s(z1)) -> c13(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), TWICE(min(0, z1)))
   F(s(0), s(z1)) -> c13(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), TWICE(min(0, z1)))

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

(254)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1


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

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

(256)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_1


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

(257) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(258)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_1


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

(259) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) by 
   F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))

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

(260)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1


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

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

(262)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1


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

(263) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(264)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1


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

(265) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) by 
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))

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

(266)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1


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

(267) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))

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

(268)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1


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

(269) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 14 trailing tuple parts
----------------------------------------

(270)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


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

(271) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) by F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
----------------------------------------

(272)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


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

(273) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) by F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
----------------------------------------

(274)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


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

(275) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) by F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
----------------------------------------

(276)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


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

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

(278)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c11_2


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

(280)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1


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

(281) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
We considered the (Usable) Rules:none
And the Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(282)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1


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

(283) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) by F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
----------------------------------------

(284)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1


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

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

(286)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c11_2


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

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

(288)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1


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

(289) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
We considered the (Usable) Rules:none
And the Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(290)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1


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

(291) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) by 
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))

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

(292)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
S tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   TWICE_1, -'_2, F_2, MIN_2, MAX_2

Compound Symbols:   c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1


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

(293) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))

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

(294)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
S tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   TWICE_1, -'_2, F_2, MIN_2, MAX_2

Compound Symbols:   c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1


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

(295) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing tuple parts
----------------------------------------

(296)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
S tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   TWICE_1, -'_2, F_2, MIN_2, MAX_2

Compound Symbols:   c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2


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

(297) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TWICE(s(z0)) -> c7(TWICE(z0)) by 
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))

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

(298)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
S tuples:
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   -'_2, F_2, MIN_2, MAX_2, TWICE_1

Compound Symbols:   c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1


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

(299) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(z0), s(z1)) -> c9(-'(z0, z1)) by 
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))

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

(300)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

(301) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) by F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
----------------------------------------

(302)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

(303) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) by F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
----------------------------------------

(304)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

(305) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) by 
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))

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

(306)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

(307) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) by none

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

(308)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

(309) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) by none

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

(310)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

(311) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(312)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


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

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

(314)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


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

(315) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = x_1 + x_2
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(316)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


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

(317) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
----------------------------------------

(318)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


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(319) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(320)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


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

(321) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = x_1 + x_2
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

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

(322)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


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

(323) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) by 
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))

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

(324)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

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

(326)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

(327) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(328)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

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

(330)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

(331) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))

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

(332)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

(333) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0)))
----------------------------------------

(334)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

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

(336)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

(337) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))

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

(338)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


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

(339) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) by 
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))

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

(340)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

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

(342)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

(343) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c13(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), TWICE(min(0, z1))) by F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(344)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

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

(346)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

(347) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c13(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), TWICE(min(0, z1))) by F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(348)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

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

(350)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

(351) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
----------------------------------------

(352)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


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

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

(354)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1