KILLED
proof of input_Q1pwKPezON.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 10 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 5132 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), 241 ms]
            (22) typed CpxTrs
(23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(24) CdtProblem
    (25) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (26) CdtProblem
    (27) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CpxRelTRS
        (31) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (32) CpxTRS
        (33) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (34) CpxWeightedTrs
            (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (36) CpxTypedWeightedTrs
            (37) CompletionProof [UPPER BOUND(ID), 0 ms]
            (38) CpxTypedWeightedCompleteTrs
                (39) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxTypedWeightedCompleteTrs
                (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
                (42) CpxRNTS
                (43) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (44) CpxRNTS
                (45) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (46) CpxRNTS
                (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (48) CpxRNTS
                (49) IntTrsBoundProof [UPPER BOUND(ID), 375 ms]
                (50) CpxRNTS
                (51) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
                (52) CpxRNTS
                (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (54) CpxRNTS
                (55) IntTrsBoundProof [UPPER BOUND(ID), 270 ms]
                (56) CpxRNTS
                (57) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
                (58) CpxRNTS
                (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (60) CpxRNTS
                (61) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
                (62) CpxRNTS
                (63) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
                (64) CpxRNTS
                (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (66) CpxRNTS
                (67) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
                (68) CpxRNTS
                (69) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
                (70) CpxRNTS
                (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (72) CpxRNTS
                (73) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
                (74) CpxRNTS
                (75) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
                (76) CpxRNTS
                (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (78) CpxRNTS
                (79) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
                (80) CpxRNTS
                (81) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
                (82) CpxRNTS
                (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (84) CpxRNTS
                (85) IntTrsBoundProof [UPPER BOUND(ID), 1990 ms]
                (86) CpxRNTS
                (87) IntTrsBoundProof [UPPER BOUND(ID), 1600 ms]
                (88) CpxRNTS
            (89) CompletionProof [UPPER BOUND(ID), 3 ms]
            (90) CpxTypedWeightedCompleteTrs
                (91) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (92) CpxRNTS
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
        (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (96) CdtProblem
        (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (98) CdtProblem
        (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
        (100) CdtProblem
        (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
        (104) CdtProblem
        (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 8 ms]
        (106) CdtProblem
        (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
        (110) CdtProblem
        (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem
        (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
        (116) CdtProblem
        (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms]
        (118) CdtProblem
        (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (120) CdtProblem
        (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CdtProblem
        (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (126) CdtProblem
        (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (130) CdtProblem
        (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (134) CdtProblem
        (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (136) CdtProblem
        (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (140) CdtProblem
        (141) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
        (142) CdtProblem
        (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (144) CdtProblem
        (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 10 ms]
        (148) CdtProblem
        (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (150) CdtProblem
        (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (152) CdtProblem
        (153) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
        (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 8 ms]
        (156) CdtProblem
        (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (158) CdtProblem
        (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (160) CdtProblem
        (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (162) CdtProblem
        (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
        (164) CdtProblem
        (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms]
        (166) CdtProblem
        (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (168) CdtProblem
        (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (170) CdtProblem
        (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (172) CdtProblem
        (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
        (174) CdtProblem
        (175) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
        (176) CdtProblem
        (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (178) CdtProblem
        (179) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CdtProblem
        (181) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CdtProblem
        (183) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
        (184) CdtProblem
        (185) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
        (186) CdtProblem
        (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (188) CdtProblem
        (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (190) CdtProblem
        (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (192) CdtProblem
        (193) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
        (194) CdtProblem
        (195) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
        (196) CdtProblem
        (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (198) CdtProblem
        (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (200) CdtProblem
        (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (202) CdtProblem
        (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (204) CdtProblem
        (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (206) CdtProblem
        (207) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (208) CdtProblem
        (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (210) CdtProblem
        (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
        (224) CdtProblem
        (225) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
        (226) CdtProblem
        (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (228) CdtProblem
        (229) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (230) CdtProblem
        (231) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (232) CdtProblem
        (233) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 10 ms]
        (234) CdtProblem
        (235) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
        (236) CdtProblem
        (237) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (238) CdtProblem
        (239) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (240) CdtProblem
        (241) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (242) CdtProblem
        (243) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 9 ms]
        (244) CdtProblem
        (245) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
        (246) CdtProblem
        (247) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (248) CdtProblem
        (249) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (250) CdtProblem
        (251) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms]
        (252) CdtProblem
        (253) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 34 ms]
        (254) CdtProblem
        (255) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (256) CdtProblem
        (257) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (258) CdtProblem
        (259) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (260) CdtProblem
        (261) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (262) CdtProblem
        (263) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (264) CdtProblem
        (265) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (266) CdtProblem
        (267) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (268) CdtProblem
        (269) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (270) CdtProblem
        (271) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (272) CdtProblem
        (273) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
        (274) CdtProblem
        (275) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (276) CdtProblem
        (277) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (278) CdtProblem
        (279) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (280) CdtProblem
        (281) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (282) CdtProblem
        (283) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 74 ms]
        (284) CdtProblem
        (285) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (286) CdtProblem
        (287) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 92 ms]
        (288) CdtProblem
        (289) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (290) CdtProblem
        (291) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (292) CdtProblem
        (293) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (294) CdtProblem
        (295) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 106 ms]
        (296) CdtProblem
        (297) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (298) CdtProblem
        (299) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (300) CdtProblem
        (301) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 101 ms]
        (302) CdtProblem
        (303) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (304) CdtProblem
(305) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(306) CpxWeightedTrs
    (307) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (308) CpxTypedWeightedTrs
    (309) CompletionProof [UPPER BOUND(ID), 0 ms]
    (310) CpxTypedWeightedCompleteTrs
        (311) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (312) CpxTypedWeightedCompleteTrs
        (313) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
        (314) CpxRNTS
        (315) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (316) CpxRNTS
        (317) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (318) CpxRNTS
        (319) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (320) CpxRNTS
        (321) IntTrsBoundProof [UPPER BOUND(ID), 325 ms]
        (322) CpxRNTS
        (323) IntTrsBoundProof [UPPER BOUND(ID), 221 ms]
        (324) CpxRNTS
        (325) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (326) CpxRNTS
        (327) IntTrsBoundProof [UPPER BOUND(ID), 113 ms]
        (328) CpxRNTS
        (329) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
        (330) CpxRNTS
        (331) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (332) CpxRNTS
        (333) IntTrsBoundProof [UPPER BOUND(ID), 3634 ms]
        (334) CpxRNTS
        (335) IntTrsBoundProof [UPPER BOUND(ID), 2585 ms]
        (336) CpxRNTS
    (337) CompletionProof [UPPER BOUND(ID), 0 ms]
    (338) CpxTypedWeightedCompleteTrs
        (339) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
        (340) CpxRNTS


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

(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:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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:

   g(x, 0') -> 0'
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0')) -> 0'
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0'), y) -> y
   id(x) -> f(x, s(0'))

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:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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:
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))
Tuples:
   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))
S tuples:
   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))
K tuples:none
Defined Rule Symbols:   g_2, double_1, half_1, f_2, id_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2, ID_1

Compound Symbols:   c, c1_1, c2, c3_1, c4_1, c5_1, c6_2, c7_2, c8, c9_1


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

(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:

   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))

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

   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

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:

   G(z0, 0') -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0')) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0'), z0) -> c8
   ID(z0) -> c9(F(z0, s(0')))

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

   g(z0, 0') -> 0'
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0')) -> 0'
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0'), z0) -> z0
   id(z0) -> f(z0, s(0'))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
G(z0, 0') -> c
G(d, s(z0)) -> c1(G(d, z0))
G(h, s(0')) -> c2
G(h, s(s(z0))) -> c3(G(h, z0))
DOUBLE(z0) -> c4(G(d, z0))
HALF(z0) -> c5(G(h, z0))
F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
F(s(0'), z0) -> c8
ID(z0) -> c9(F(z0, s(0')))
g(z0, 0') -> 0'
g(d, s(z0)) -> s(s(g(d, z0)))
g(h, s(0')) -> 0'
g(h, s(s(z0))) -> s(g(h, z0))
double(z0) -> g(d, z0)
half(z0) -> g(h, z0)
f(s(z0), z1) -> f(half(s(z0)), double(z1))
f(s(0'), z0) -> z0
id(z0) -> f(z0, s(0'))

Types:
G :: d:h -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
d :: d:h
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
h :: d:h
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
DOUBLE :: 0':s -> c4
c4 :: c:c1:c2:c3 -> c4
HALF :: 0':s -> c5
c5 :: c:c1:c2:c3 -> c5
F :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8 -> c5 -> c6:c7:c8
half :: 0':s -> 0':s
double :: 0':s -> 0':s
c7 :: c6:c7:c8 -> c4 -> c6:c7:c8
c8 :: c6:c7:c8
ID :: 0':s -> c9
c9 :: c6:c7:c8 -> c9
g :: d:h -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_c:c1:c2:c31_10 :: c:c1:c2:c3
hole_d:h2_10 :: d:h
hole_0':s3_10 :: 0':s
hole_c44_10 :: c4
hole_c55_10 :: c5
hole_c6:c7:c86_10 :: c6:c7:c8
hole_c97_10 :: c9
gen_c:c1:c2:c38_10 :: Nat -> c:c1:c2:c3
gen_0':s9_10 :: Nat -> 0':s
gen_c6:c7:c810_10 :: Nat -> c6:c7:c8

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
G, F, g, f
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
G(z0, 0') -> c
G(d, s(z0)) -> c1(G(d, z0))
G(h, s(0')) -> c2
G(h, s(s(z0))) -> c3(G(h, z0))
DOUBLE(z0) -> c4(G(d, z0))
HALF(z0) -> c5(G(h, z0))
F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
F(s(0'), z0) -> c8
ID(z0) -> c9(F(z0, s(0')))
g(z0, 0') -> 0'
g(d, s(z0)) -> s(s(g(d, z0)))
g(h, s(0')) -> 0'
g(h, s(s(z0))) -> s(g(h, z0))
double(z0) -> g(d, z0)
half(z0) -> g(h, z0)
f(s(z0), z1) -> f(half(s(z0)), double(z1))
f(s(0'), z0) -> z0
id(z0) -> f(z0, s(0'))

Types:
G :: d:h -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
d :: d:h
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
h :: d:h
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
DOUBLE :: 0':s -> c4
c4 :: c:c1:c2:c3 -> c4
HALF :: 0':s -> c5
c5 :: c:c1:c2:c3 -> c5
F :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8 -> c5 -> c6:c7:c8
half :: 0':s -> 0':s
double :: 0':s -> 0':s
c7 :: c6:c7:c8 -> c4 -> c6:c7:c8
c8 :: c6:c7:c8
ID :: 0':s -> c9
c9 :: c6:c7:c8 -> c9
g :: d:h -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_c:c1:c2:c31_10 :: c:c1:c2:c3
hole_d:h2_10 :: d:h
hole_0':s3_10 :: 0':s
hole_c44_10 :: c4
hole_c55_10 :: c5
hole_c6:c7:c86_10 :: c6:c7:c8
hole_c97_10 :: c9
gen_c:c1:c2:c38_10 :: Nat -> c:c1:c2:c3
gen_0':s9_10 :: Nat -> 0':s
gen_c6:c7:c810_10 :: Nat -> c6:c7:c8


Generator Equations:
gen_c:c1:c2:c38_10(0) <=> c
gen_c:c1:c2:c38_10(+(x, 1)) <=> c1(gen_c:c1:c2:c38_10(x))
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c6:c7:c810_10(0) <=> c8
gen_c6:c7:c810_10(+(x, 1)) <=> c6(gen_c6:c7:c810_10(x), c5(c))


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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
G(h, gen_0':s9_10(+(2, *(2, n12_10)))) -> *11_10, rt in Omega(n12_10)

Induction Base:
G(h, gen_0':s9_10(+(2, *(2, 0))))

Induction Step:
G(h, gen_0':s9_10(+(2, *(2, +(n12_10, 1))))) ->_R^Omega(1)
c3(G(h, gen_0':s9_10(+(2, *(2, n12_10))))) ->_IH
c3(*11_10)

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:
G(z0, 0') -> c
G(d, s(z0)) -> c1(G(d, z0))
G(h, s(0')) -> c2
G(h, s(s(z0))) -> c3(G(h, z0))
DOUBLE(z0) -> c4(G(d, z0))
HALF(z0) -> c5(G(h, z0))
F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
F(s(0'), z0) -> c8
ID(z0) -> c9(F(z0, s(0')))
g(z0, 0') -> 0'
g(d, s(z0)) -> s(s(g(d, z0)))
g(h, s(0')) -> 0'
g(h, s(s(z0))) -> s(g(h, z0))
double(z0) -> g(d, z0)
half(z0) -> g(h, z0)
f(s(z0), z1) -> f(half(s(z0)), double(z1))
f(s(0'), z0) -> z0
id(z0) -> f(z0, s(0'))

Types:
G :: d:h -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
d :: d:h
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
h :: d:h
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
DOUBLE :: 0':s -> c4
c4 :: c:c1:c2:c3 -> c4
HALF :: 0':s -> c5
c5 :: c:c1:c2:c3 -> c5
F :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8 -> c5 -> c6:c7:c8
half :: 0':s -> 0':s
double :: 0':s -> 0':s
c7 :: c6:c7:c8 -> c4 -> c6:c7:c8
c8 :: c6:c7:c8
ID :: 0':s -> c9
c9 :: c6:c7:c8 -> c9
g :: d:h -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_c:c1:c2:c31_10 :: c:c1:c2:c3
hole_d:h2_10 :: d:h
hole_0':s3_10 :: 0':s
hole_c44_10 :: c4
hole_c55_10 :: c5
hole_c6:c7:c86_10 :: c6:c7:c8
hole_c97_10 :: c9
gen_c:c1:c2:c38_10 :: Nat -> c:c1:c2:c3
gen_0':s9_10 :: Nat -> 0':s
gen_c6:c7:c810_10 :: Nat -> c6:c7:c8


Generator Equations:
gen_c:c1:c2:c38_10(0) <=> c
gen_c:c1:c2:c38_10(+(x, 1)) <=> c1(gen_c:c1:c2:c38_10(x))
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c6:c7:c810_10(0) <=> c8
gen_c6:c7:c810_10(+(x, 1)) <=> c6(gen_c6:c7:c810_10(x), c5(c))


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

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
G(z0, 0') -> c
G(d, s(z0)) -> c1(G(d, z0))
G(h, s(0')) -> c2
G(h, s(s(z0))) -> c3(G(h, z0))
DOUBLE(z0) -> c4(G(d, z0))
HALF(z0) -> c5(G(h, z0))
F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
F(s(0'), z0) -> c8
ID(z0) -> c9(F(z0, s(0')))
g(z0, 0') -> 0'
g(d, s(z0)) -> s(s(g(d, z0)))
g(h, s(0')) -> 0'
g(h, s(s(z0))) -> s(g(h, z0))
double(z0) -> g(d, z0)
half(z0) -> g(h, z0)
f(s(z0), z1) -> f(half(s(z0)), double(z1))
f(s(0'), z0) -> z0
id(z0) -> f(z0, s(0'))

Types:
G :: d:h -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
d :: d:h
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
h :: d:h
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
DOUBLE :: 0':s -> c4
c4 :: c:c1:c2:c3 -> c4
HALF :: 0':s -> c5
c5 :: c:c1:c2:c3 -> c5
F :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8 -> c5 -> c6:c7:c8
half :: 0':s -> 0':s
double :: 0':s -> 0':s
c7 :: c6:c7:c8 -> c4 -> c6:c7:c8
c8 :: c6:c7:c8
ID :: 0':s -> c9
c9 :: c6:c7:c8 -> c9
g :: d:h -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_c:c1:c2:c31_10 :: c:c1:c2:c3
hole_d:h2_10 :: d:h
hole_0':s3_10 :: 0':s
hole_c44_10 :: c4
hole_c55_10 :: c5
hole_c6:c7:c86_10 :: c6:c7:c8
hole_c97_10 :: c9
gen_c:c1:c2:c38_10 :: Nat -> c:c1:c2:c3
gen_0':s9_10 :: Nat -> 0':s
gen_c6:c7:c810_10 :: Nat -> c6:c7:c8


Lemmas:
G(h, gen_0':s9_10(+(2, *(2, n12_10)))) -> *11_10, rt in Omega(n12_10)


Generator Equations:
gen_c:c1:c2:c38_10(0) <=> c
gen_c:c1:c2:c38_10(+(x, 1)) <=> c1(gen_c:c1:c2:c38_10(x))
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c6:c7:c810_10(0) <=> c8
gen_c6:c7:c810_10(+(x, 1)) <=> c6(gen_c6:c7:c810_10(x), c5(c))


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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(h, gen_0':s9_10(*(2, n5577_10))) -> gen_0':s9_10(n5577_10), rt in Omega(0)

Induction Base:
g(h, gen_0':s9_10(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
g(h, gen_0':s9_10(*(2, +(n5577_10, 1)))) ->_R^Omega(0)
s(g(h, gen_0':s9_10(*(2, n5577_10)))) ->_IH
s(gen_0':s9_10(c5578_10))

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

(22)
Obligation:
Innermost TRS:
Rules:
G(z0, 0') -> c
G(d, s(z0)) -> c1(G(d, z0))
G(h, s(0')) -> c2
G(h, s(s(z0))) -> c3(G(h, z0))
DOUBLE(z0) -> c4(G(d, z0))
HALF(z0) -> c5(G(h, z0))
F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
F(s(0'), z0) -> c8
ID(z0) -> c9(F(z0, s(0')))
g(z0, 0') -> 0'
g(d, s(z0)) -> s(s(g(d, z0)))
g(h, s(0')) -> 0'
g(h, s(s(z0))) -> s(g(h, z0))
double(z0) -> g(d, z0)
half(z0) -> g(h, z0)
f(s(z0), z1) -> f(half(s(z0)), double(z1))
f(s(0'), z0) -> z0
id(z0) -> f(z0, s(0'))

Types:
G :: d:h -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
d :: d:h
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
h :: d:h
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
DOUBLE :: 0':s -> c4
c4 :: c:c1:c2:c3 -> c4
HALF :: 0':s -> c5
c5 :: c:c1:c2:c3 -> c5
F :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8 -> c5 -> c6:c7:c8
half :: 0':s -> 0':s
double :: 0':s -> 0':s
c7 :: c6:c7:c8 -> c4 -> c6:c7:c8
c8 :: c6:c7:c8
ID :: 0':s -> c9
c9 :: c6:c7:c8 -> c9
g :: d:h -> 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_c:c1:c2:c31_10 :: c:c1:c2:c3
hole_d:h2_10 :: d:h
hole_0':s3_10 :: 0':s
hole_c44_10 :: c4
hole_c55_10 :: c5
hole_c6:c7:c86_10 :: c6:c7:c8
hole_c97_10 :: c9
gen_c:c1:c2:c38_10 :: Nat -> c:c1:c2:c3
gen_0':s9_10 :: Nat -> 0':s
gen_c6:c7:c810_10 :: Nat -> c6:c7:c8


Lemmas:
G(h, gen_0':s9_10(+(2, *(2, n12_10)))) -> *11_10, rt in Omega(n12_10)
g(h, gen_0':s9_10(*(2, n5577_10))) -> gen_0':s9_10(n5577_10), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c38_10(0) <=> c
gen_c:c1:c2:c38_10(+(x, 1)) <=> c1(gen_c:c1:c2:c38_10(x))
gen_0':s9_10(0) <=> 0'
gen_0':s9_10(+(x, 1)) <=> s(gen_0':s9_10(x))
gen_c6:c7:c810_10(0) <=> c8
gen_c6:c7:c810_10(+(x, 1)) <=> c6(gen_c6:c7:c810_10(x), c5(c))


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

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))
Tuples:
   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))
S tuples:
   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))
K tuples:none
Defined Rule Symbols:   g_2, double_1, half_1, f_2, id_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2, ID_1

Compound Symbols:   c, c1_1, c2, c3_1, c4_1, c5_1, c6_2, c7_2, c8, c9_1


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

(25) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ID(z0) -> c9(F(z0, s(0)))
Removed 3 trailing nodes:
   F(s(0), z0) -> c8
   G(z0, 0) -> c
   G(h, s(0)) -> c2

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
K tuples:none
Defined Rule Symbols:   g_2, double_1, half_1, f_2, id_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c7_2


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

(27) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
K tuples:none
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c7_2


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

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

(30)
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:

   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))

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

   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)

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

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

(32)
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:

   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)

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

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

(34)
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:

   G(d, s(z0)) -> c1(G(d, z0)) [1]
   G(h, s(s(z0))) -> c3(G(h, z0)) [1]
   DOUBLE(z0) -> c4(G(d, z0)) [1]
   HALF(z0) -> c5(G(h, z0)) [1]
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0))) [1]
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1)) [1]
   half(z0) -> g(h, z0) [0]
   g(z0, 0) -> 0 [0]
   g(h, s(0)) -> 0 [0]
   g(h, s(s(z0))) -> s(g(h, z0)) [0]
   g(d, s(z0)) -> s(s(g(d, z0))) [0]
   double(z0) -> g(d, z0) [0]

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

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

(36)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   G(d, s(z0)) -> c1(G(d, z0)) [1]
   G(h, s(s(z0))) -> c3(G(h, z0)) [1]
   DOUBLE(z0) -> c4(G(d, z0)) [1]
   HALF(z0) -> c5(G(h, z0)) [1]
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0))) [1]
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1)) [1]
   half(z0) -> g(h, z0) [0]
   g(z0, 0) -> 0 [0]
   g(h, s(0)) -> 0 [0]
   g(h, s(s(z0))) -> s(g(h, z0)) [0]
   g(d, s(z0)) -> s(s(g(d, z0))) [0]
   double(z0) -> g(d, z0) [0]

The TRS has the following type information:
G :: d:h -> s:0 -> c1:c3
d :: d:h
s :: s:0 -> s:0
c1 :: c1:c3 -> c1:c3
h :: d:h
c3 :: c1:c3 -> c1:c3
DOUBLE :: s:0 -> c4
c4 :: c1:c3 -> c4
HALF :: s:0 -> c5
c5 :: c1:c3 -> c5
F :: s:0 -> s:0 -> c6:c7
c6 :: c6:c7 -> c5 -> c6:c7
half :: s:0 -> s:0
double :: s:0 -> s:0
c7 :: c6:c7 -> c4 -> c6:c7
g :: d:h -> s:0 -> s:0
0 :: s:0

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

(37) 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:

   G_2
   DOUBLE_1
   HALF_1
   F_2

(c) The following functions are completely defined:

   half_1
   g_2
   double_1

Due to the following rules being added:

   half(v0) -> 0 [0]
   g(v0, v1) -> 0 [0]
   double(v0) -> 0 [0]

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

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

(38)
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:

   G(d, s(z0)) -> c1(G(d, z0)) [1]
   G(h, s(s(z0))) -> c3(G(h, z0)) [1]
   DOUBLE(z0) -> c4(G(d, z0)) [1]
   HALF(z0) -> c5(G(h, z0)) [1]
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0))) [1]
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1)) [1]
   half(z0) -> g(h, z0) [0]
   g(z0, 0) -> 0 [0]
   g(h, s(0)) -> 0 [0]
   g(h, s(s(z0))) -> s(g(h, z0)) [0]
   g(d, s(z0)) -> s(s(g(d, z0))) [0]
   double(z0) -> g(d, z0) [0]
   half(v0) -> 0 [0]
   g(v0, v1) -> 0 [0]
   double(v0) -> 0 [0]

The TRS has the following type information:
G :: d:h -> s:0 -> c1:c3
d :: d:h
s :: s:0 -> s:0
c1 :: c1:c3 -> c1:c3
h :: d:h
c3 :: c1:c3 -> c1:c3
DOUBLE :: s:0 -> c4
c4 :: c1:c3 -> c4
HALF :: s:0 -> c5
c5 :: c1:c3 -> c5
F :: s:0 -> s:0 -> c6:c7
c6 :: c6:c7 -> c5 -> c6:c7
half :: s:0 -> s:0
double :: s:0 -> s:0
c7 :: c6:c7 -> c4 -> c6:c7
g :: d:h -> s:0 -> s:0
0 :: s:0
const :: c1:c3
const1 :: c4
const2 :: c5
const3 :: c6:c7

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

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

(40)
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:

   G(d, s(z0)) -> c1(G(d, z0)) [1]
   G(h, s(s(z0))) -> c3(G(h, z0)) [1]
   DOUBLE(z0) -> c4(G(d, z0)) [1]
   HALF(z0) -> c5(G(h, z0)) [1]
   F(s(z0), z1) -> c6(F(g(h, s(z0)), g(d, z1)), HALF(s(z0))) [1]
   F(s(z0), z1) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) [1]
   F(s(z0), z1) -> c6(F(0, g(d, z1)), HALF(s(z0))) [1]
   F(s(z0), z1) -> c6(F(0, 0), HALF(s(z0))) [1]
   F(s(z0), z1) -> c7(F(g(h, s(z0)), g(d, z1)), DOUBLE(z1)) [1]
   F(s(z0), z1) -> c7(F(g(h, s(z0)), 0), DOUBLE(z1)) [1]
   F(s(z0), z1) -> c7(F(0, g(d, z1)), DOUBLE(z1)) [1]
   F(s(z0), z1) -> c7(F(0, 0), DOUBLE(z1)) [1]
   half(z0) -> g(h, z0) [0]
   g(z0, 0) -> 0 [0]
   g(h, s(0)) -> 0 [0]
   g(h, s(s(z0))) -> s(g(h, z0)) [0]
   g(d, s(z0)) -> s(s(g(d, z0))) [0]
   double(z0) -> g(d, z0) [0]
   half(v0) -> 0 [0]
   g(v0, v1) -> 0 [0]
   double(v0) -> 0 [0]

The TRS has the following type information:
G :: d:h -> s:0 -> c1:c3
d :: d:h
s :: s:0 -> s:0
c1 :: c1:c3 -> c1:c3
h :: d:h
c3 :: c1:c3 -> c1:c3
DOUBLE :: s:0 -> c4
c4 :: c1:c3 -> c4
HALF :: s:0 -> c5
c5 :: c1:c3 -> c5
F :: s:0 -> s:0 -> c6:c7
c6 :: c6:c7 -> c5 -> c6:c7
half :: s:0 -> s:0
double :: s:0 -> s:0
c7 :: c6:c7 -> c4 -> c6:c7
g :: d:h -> s:0 -> s:0
0 :: s:0
const :: c1:c3
const1 :: c4
const2 :: c5
const3 :: c6:c7

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

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

   d => 0
   h => 1
   0 => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z0) :|: z = z0, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + z0), g(0, z1)) + HALF(1 + z0) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + z0), g(0, z1)) + DOUBLE(z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + z0), 0) + HALF(1 + z0) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + z0), 0) + DOUBLE(z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z1)) + HALF(1 + z0) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z1)) + DOUBLE(z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + z0) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z0) :|: z = 1, z0 >= 0, z' = 1 + (1 + z0)
   G(z, z') -{ 1 }-> 1 + G(0, z0) :|: z0 >= 0, z' = 1 + z0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z0) :|: z = z0, z0 >= 0
   double(z) -{ 0 }-> g(0, z0) :|: z = z0, z0 >= 0
   double(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 0 }-> 1 + g(1, z0) :|: z = 1, z0 >= 0, z' = 1 + (1 + z0)
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z0)) :|: z0 >= 0, z' = 1 + z0, z = 0
   half(z) -{ 0 }-> g(1, z0) :|: z = z0, z0 >= 0
   half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> g(0, z) :|: z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> g(1, z) :|: z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { g }
   { G }
   { double }
   { half }
   { HALF }
   { DOUBLE }
   { F }

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> g(0, z) :|: z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> g(1, z) :|: z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {G}, {double}, {half}, {HALF}, {DOUBLE}, {F}

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

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> g(0, z) :|: z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> g(1, z) :|: z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {G}, {double}, {half}, {HALF}, {DOUBLE}, {F}

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> g(0, z) :|: z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> g(1, z) :|: z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {G}, {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: ?, size: O(n^1) [2*z']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(g(1, 1 + (z - 1)), 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, g(0, z')) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> g(0, z) :|: z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> g(1, z) :|: z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {G}, {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']

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

(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:

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {G}, {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {G}, {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: ?, size: O(1) [0]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: G
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:

   DOUBLE(z) -{ 1 }-> 1 + G(0, z) :|: z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + G(1, z' - 2) :|: z = 1, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + G(0, z' - 1) :|: z' - 1 >= 0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z) :|: z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]

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

(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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {double}, {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: ?, size: O(n^1) [2*z]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {half}, {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: ?, size: O(n^1) [2*z]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {HALF}, {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: ?, size: O(1) [1]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: HALF
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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 1 }-> 1 + F(s, s') + HALF(1 + (z - 1)) :|: s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s'', 0) + HALF(1 + (z - 1)) :|: s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s1) + HALF(1 + (z - 1)) :|: s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + HALF(1 + (z - 1)) :|: z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]

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

(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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s, s') + s14 :|: s14 >= 0, s14 <= 1, s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s'', 0) + s15 :|: s15 >= 0, s15 <= 1, s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, s1) + s16 :|: s16 >= 0, s16 <= 1, s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, 0) + s17 :|: s17 >= 0, s17 <= 1, z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]

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

(79) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s, s') + s14 :|: s14 >= 0, s14 <= 1, s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s'', 0) + s15 :|: s15 >= 0, s15 <= 1, s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, s1) + s16 :|: s16 >= 0, s16 <= 1, s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, 0) + s17 :|: s17 >= 0, s17 <= 1, z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {DOUBLE}, {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]
  DOUBLE: runtime: ?, size: O(1) [1]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s, s') + s14 :|: s14 >= 0, s14 <= 1, s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s'', 0) + s15 :|: s15 >= 0, s15 <= 1, s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s2, s3) + DOUBLE(z') :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(s4, 0) + DOUBLE(z') :|: s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, s1) + s16 :|: s16 >= 0, s16 <= 1, s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, s5) + DOUBLE(z') :|: s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, 0) + s17 :|: s17 >= 0, s17 <= 1, z' >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(0, 0) + DOUBLE(z') :|: z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]
  DOUBLE: runtime: O(n^1) [2 + z], size: O(1) [1]

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

(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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s, s') + s14 :|: s14 >= 0, s14 <= 1, s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s'', 0) + s15 :|: s15 >= 0, s15 <= 1, s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(s2, s3) + s18 :|: s18 >= 0, s18 <= 1, s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(s4, 0) + s19 :|: s19 >= 0, s19 <= 1, s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, s1) + s16 :|: s16 >= 0, s16 <= 1, s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(0, s5) + s20 :|: s20 >= 0, s20 <= 1, s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, 0) + s17 :|: s17 >= 0, s17 <= 1, z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(0, 0) + s21 :|: s21 >= 0, s21 <= 1, z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]
  DOUBLE: runtime: O(n^1) [2 + z], size: O(1) [1]

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

(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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s, s') + s14 :|: s14 >= 0, s14 <= 1, s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s'', 0) + s15 :|: s15 >= 0, s15 <= 1, s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(s2, s3) + s18 :|: s18 >= 0, s18 <= 1, s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(s4, 0) + s19 :|: s19 >= 0, s19 <= 1, s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, s1) + s16 :|: s16 >= 0, s16 <= 1, s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(0, s5) + s20 :|: s20 >= 0, s20 <= 1, s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, 0) + s17 :|: s17 >= 0, s17 <= 1, z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(0, 0) + s21 :|: s21 >= 0, s21 <= 1, z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]
  DOUBLE: runtime: O(n^1) [2 + z], size: O(1) [1]
  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:

   DOUBLE(z) -{ 2 + z }-> 1 + s12 :|: s12 >= 0, s12 <= 0, z >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s, s') + s14 :|: s14 >= 0, s14 <= 1, s >= 0, s <= 2 * (1 + (z - 1)), s' >= 0, s' <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(s'', 0) + s15 :|: s15 >= 0, s15 <= 1, s'' >= 0, s'' <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(s2, s3) + s18 :|: s18 >= 0, s18 <= 1, s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(s4, 0) + s19 :|: s19 >= 0, s19 <= 1, s4 >= 0, s4 <= 2 * (1 + (z - 1)), z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, s1) + s16 :|: s16 >= 0, s16 <= 1, s1 >= 0, s1 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(0, s5) + s20 :|: s20 >= 0, s20 <= 1, s5 >= 0, s5 <= 2 * z', z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z }-> 1 + F(0, 0) + s17 :|: s17 >= 0, s17 <= 1, z' >= 0, z - 1 >= 0
   F(z, z') -{ 3 + z' }-> 1 + F(0, 0) + s21 :|: s21 >= 0, s21 <= 1, z' >= 0, z - 1 >= 0
   G(z, z') -{ 1 + z' }-> 1 + s10 :|: s10 >= 0, s10 <= 0, z' - 1 >= 0, z = 0
   G(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z = 1, z' - 2 >= 0
   HALF(z) -{ 2 + z }-> 1 + s13 :|: s13 >= 0, s13 <= 0, z >= 0
   double(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * z, z >= 0
   double(z) -{ 0 }-> 0 :|: z >= 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> 1 + s7 :|: s7 >= 0, s7 <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 0 }-> 1 + (1 + s8) :|: s8 >= 0, s8 <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 2 * z, z >= 0
   half(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  g: runtime: O(1) [0], size: O(n^1) [2*z']
  G: runtime: O(n^1) [1 + z'], size: O(1) [0]
  double: runtime: O(1) [0], size: O(n^1) [2*z]
  half: runtime: O(1) [0], size: O(n^1) [2*z]
  HALF: runtime: O(n^1) [2 + z], size: O(1) [1]
  DOUBLE: runtime: O(n^1) [2 + z], size: O(1) [1]
  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:

   half(v0) -> null_half [0]
   g(v0, v1) -> null_g [0]
   double(v0) -> null_double [0]
   G(v0, v1) -> null_G [0]
   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_half, null_g, null_double, null_G, null_F, const, const1

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

(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:

   G(d, s(z0)) -> c1(G(d, z0)) [1]
   G(h, s(s(z0))) -> c3(G(h, z0)) [1]
   DOUBLE(z0) -> c4(G(d, z0)) [1]
   HALF(z0) -> c5(G(h, z0)) [1]
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0))) [1]
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1)) [1]
   half(z0) -> g(h, z0) [0]
   g(z0, 0) -> 0 [0]
   g(h, s(0)) -> 0 [0]
   g(h, s(s(z0))) -> s(g(h, z0)) [0]
   g(d, s(z0)) -> s(s(g(d, z0))) [0]
   double(z0) -> g(d, z0) [0]
   half(v0) -> null_half [0]
   g(v0, v1) -> null_g [0]
   double(v0) -> null_double [0]
   G(v0, v1) -> null_G [0]
   F(v0, v1) -> null_F [0]

The TRS has the following type information:
G :: d:h -> s:0:null_half:null_g:null_double -> c1:c3:null_G
d :: d:h
s :: s:0:null_half:null_g:null_double -> s:0:null_half:null_g:null_double
c1 :: c1:c3:null_G -> c1:c3:null_G
h :: d:h
c3 :: c1:c3:null_G -> c1:c3:null_G
DOUBLE :: s:0:null_half:null_g:null_double -> c4
c4 :: c1:c3:null_G -> c4
HALF :: s:0:null_half:null_g:null_double -> c5
c5 :: c1:c3:null_G -> c5
F :: s:0:null_half:null_g:null_double -> s:0:null_half:null_g:null_double -> c6:c7:null_F
c6 :: c6:c7:null_F -> c5 -> c6:c7:null_F
half :: s:0:null_half:null_g:null_double -> s:0:null_half:null_g:null_double
double :: s:0:null_half:null_g:null_double -> s:0:null_half:null_g:null_double
c7 :: c6:c7:null_F -> c4 -> c6:c7:null_F
g :: d:h -> s:0:null_half:null_g:null_double -> s:0:null_half:null_g:null_double
0 :: s:0:null_half:null_g:null_double
null_half :: s:0:null_half:null_g:null_double
null_g :: s:0:null_half:null_g:null_double
null_double :: s:0:null_half:null_g:null_double
null_G :: c1:c3:null_G
null_F :: c6:c7:null_F
const :: c4
const1 :: c5

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:

   d => 0
   h => 1
   0 => 0
   null_half => 0
   null_g => 0
   null_double => 0
   null_G => 0
   null_F => 0
   const => 0
   const1 => 0

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

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

   DOUBLE(z) -{ 1 }-> 1 + G(0, z0) :|: z = z0, z0 >= 0
   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + F(half(1 + z0), double(z1)) + HALF(1 + z0) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z, z') -{ 1 }-> 1 + F(half(1 + z0), double(z1)) + DOUBLE(z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   G(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   G(z, z') -{ 1 }-> 1 + G(1, z0) :|: z = 1, z0 >= 0, z' = 1 + (1 + z0)
   G(z, z') -{ 1 }-> 1 + G(0, z0) :|: z0 >= 0, z' = 1 + z0, z = 0
   HALF(z) -{ 1 }-> 1 + G(1, z0) :|: z = z0, z0 >= 0
   double(z) -{ 0 }-> g(0, z0) :|: z = z0, z0 >= 0
   double(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 0 }-> 1 + g(1, z0) :|: z = 1, z0 >= 0, z' = 1 + (1 + z0)
   g(z, z') -{ 0 }-> 1 + (1 + g(0, z0)) :|: z0 >= 0, z' = 1 + z0, z = 0
   half(z) -{ 0 }-> g(1, z0) :|: z = z0, z0 >= 0
   half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0))) by 
   F(s(x0), z0) -> c6(F(half(s(x0)), g(d, z0)), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(x0), z0) -> c6(F(half(s(x0)), g(d, z0)), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(x0), z0) -> c6(F(half(s(x0)), g(d, z0)), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))
K tuples:none
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1)) by 
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c6(F(half(s(x0)), g(d, z0)), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c6(F(half(s(x0)), g(d, z0)), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
K tuples:none
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c7_2


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), z0) -> c6(F(half(s(x0)), g(d, z0)), HALF(s(x0))) by 
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0)))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
K tuples:none
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c7_2


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), x1) -> c6(F(g(h, s(x0)), double(x1)), HALF(s(x0))) by 
   F(s(x0), z0) -> c6(F(g(h, s(x0)), g(d, z0)), HALF(s(x0)))
   F(s(0), x1) -> c6(F(0, double(x1)), HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(0), x1) -> c6(F(0, double(x1)), HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(0), x1) -> c6(F(0, double(x1)), HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
K tuples:none
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
K tuples:none
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

(103) 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), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = 0
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = [1]
   POL(half(x_1)) = [1] + x_1
   POL(s(x_1)) = 0

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

(105) 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(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   half(z0) -> g(h, z0)
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(half(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), z0) -> c7(F(half(s(x0)), g(d, z0)), DOUBLE(z0)) by 
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1, c7_1


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

(109) 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), x1) -> c7(DOUBLE(x1))
We considered the (Usable) Rules:none
And the Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = 0
   POL(HALF(x_1)) = x_1
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = [1]
   POL(half(x_1)) = [1] + x_1
   POL(s(x_1)) = 0

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1, c7_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), x1) -> c7(F(g(h, s(x0)), double(x1)), DOUBLE(x1)) by 
   F(s(x0), z0) -> c7(F(g(h, s(x0)), g(d, z0)), DOUBLE(z0))
   F(s(0), x1) -> c7(F(0, double(x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(F(0, double(x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(F(0, double(x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(115) 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), x1) -> c7(DOUBLE(x1))
We considered the (Usable) Rules:none
And the Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = 0
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [3] + [3]x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = 0
   POL(half(x_1)) = [3] + [3]x_1
   POL(s(x_1)) = 0

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(117) 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(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   half(z0) -> g(h, z0)
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(half(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(119) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0), 0) -> c6(F(half(s(x0)), 0), HALF(s(x0))) by F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0), s(z0)) -> c6(F(half(s(x0)), s(s(g(d, z0)))), HALF(s(x0))) by F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0)))) by F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0), 0) -> c7(F(half(s(x0)), 0), DOUBLE(0)) by F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
S tuples:
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c1_1, c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1


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

(127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(d, s(z0)) -> c1(G(d, z0)) by 
   G(d, s(s(y0))) -> c1(G(d, s(y0)))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
S tuples:
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   G_2, DOUBLE_1, HALF_1, F_2

Compound Symbols:   c3_1, c4_1, c5_1, c6_2, c6_1, c7_2, c7_1, c1_1


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

(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(h, s(s(z0))) -> c3(G(h, z0)) by 
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
S tuples:
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, F_2, G_2

Compound Symbols:   c4_1, c5_1, c6_2, c6_1, c7_2, c7_1, c1_1, c3_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DOUBLE(z0) -> c4(G(d, z0)) by 
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
S tuples:
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0), DOUBLE(0))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   HALF_1, F_2, G_2, DOUBLE_1

Compound Symbols:   c5_1, c6_2, c6_1, c7_2, c7_1, c1_1, c3_1, c4_1


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

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
S tuples:
   HALF(z0) -> c5(G(h, z0))
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   HALF_1, F_2, G_2, DOUBLE_1

Compound Symbols:   c5_1, c6_2, c6_1, c7_2, c7_1, c1_1, c3_1, c4_1


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

(135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace HALF(z0) -> c5(G(h, z0)) by 
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
S tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(0), x1) -> c6(HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), double(x1)), HALF(s(s(z0))))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_2, c6_1, c7_2, c7_1, c1_1, c3_1, c4_1, c5_1


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

(137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), x1) -> c6(HALF(s(0)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
S tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_2, c6_1, c7_2, c7_1, c1_1, c3_1, c4_1, c5_1


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

(139) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0), s(z0)) -> c7(F(half(s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0))) by F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(z0) -> g(h, z0)
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
S tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   half_1, g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_2, c6_1, c7_2, c7_1, c1_1, c3_1, c4_1, c5_1


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

(141) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   half(z0) -> g(h, z0)

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
S tuples:
   F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_2, c6_1, c7_2, c7_1, c1_1, c3_1, c4_1, c5_1


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), x1) -> c6(F(g(h, s(x0)), g(d, x1)), HALF(s(x0))) by 
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(0), x1) -> c6(F(0, g(d, x1)), HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), g(d, x1)), HALF(s(s(z0))))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(0), x1) -> c6(F(0, g(d, x1)), HALF(s(0)))
S tuples:
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(0), x1) -> c6(F(0, g(d, x1)), HALF(s(0)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), g(d, x1)), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_2, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1


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

(145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), x1) -> c6(F(0, g(d, x1)), HALF(s(0)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
S tuples:
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), g(d, x1)), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_2, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1


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

(147) 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(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
S tuples:
   F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_2, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1


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

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), x1) -> c7(F(g(h, s(x0)), g(d, x1)), DOUBLE(x1)) by 
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0), DOUBLE(0))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(0), x1) -> c7(F(0, g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0), DOUBLE(0))
   F(s(0), x1) -> c7(F(0, g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
S tuples:
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0), DOUBLE(0))
   F(s(0), x1) -> c7(F(0, g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c7_2, c6_2, c1_1, c3_1, c4_1, c5_1


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

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
S tuples:
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c7_2, c6_2, c1_1, c3_1, c4_1, c5_1


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

(153) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
S tuples:
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c7_2, c6_2, c1_1, c3_1, c4_1, c5_1


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

(155) 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(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
S tuples:
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c7_2, c6_2, c1_1, c3_1, c4_1, c5_1


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

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) by 
   F(s(s(x0)), z0) -> c7(F(s(g(h, x0)), g(d, z0)), DOUBLE(z0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
S tuples:
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(159) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) by 
   F(s(0), 0) -> c6(F(0, 0), HALF(s(0)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(0), 0) -> c6(F(0, 0), HALF(s(0)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
S tuples:
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(0), 0) -> c6(F(0, 0), HALF(s(0)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
S tuples:
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(163) 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), 0) -> c6
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = [3]x_2
   POL(HALF(x_1)) = [3]x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [3] + [3]x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = 0
   POL(s(x_1)) = 0

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
S tuples:
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(0), 0) -> c6
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(165) 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(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
S tuples:
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(0), 0) -> c6
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) by 
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0)))
   F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0)))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), 0) -> c6
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(0), 0) -> c6
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), 0) -> c6

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(173) 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(x1)) -> c6
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = x_2
   POL(HALF(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = [1]
   POL(s(x_1)) = 0

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(175) 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(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), 0) -> c7(F(g(h, s(z0)), 0)) by 
   F(s(0), 0) -> c7(F(0, 0))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   F(s(z0), 0) -> c7(F(g(h, s(z0)), 0))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
   F(s(0), 0) -> c7(F(0, 0))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), 0) -> c7(F(0, 0))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(0), s(x1)) -> c6
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(179) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(x1)) -> c6

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), 0) -> c7(F(0, 0))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), 0) -> c7(F(0, 0))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(183) 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), 0) -> c7
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = x_1 + x_2
   POL(HALF(x_1)) = x_1
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = 0
   POL(s(x_1)) = 0

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), 0) -> c7
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(185) 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(z0)), 0) -> c7(F(s(g(h, z0)), 0))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), 0) -> c7
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(187) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) by 
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1)))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), 0) -> c7
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(0), 0) -> c7
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), 0) -> c7

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(193) 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(x1)) -> c7(DOUBLE(s(x1)))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [2]
   POL(G(x_1, x_2)) = [3]x_2
   POL(HALF(x_1)) = [3]x_1
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [3] + [3]x_1
   POL(g(x_1, x_2)) = [1] + [3]x_1
   POL(h) = [3]
   POL(s(x_1)) = 0

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(195) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0)))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(197) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), 0) -> c6(F(g(h, s(x0)), 0), HALF(s(x0))) by 
   F(s(0), 0) -> c6(F(0, 0), HALF(s(0)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(0), 0) -> c6(F(0, 0), HALF(s(0)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), 0) -> c6(F(0, 0), HALF(s(0)))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0)))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(201) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), s(z0)) -> c6(F(g(h, s(x0)), s(s(g(d, z0)))), HALF(s(x0))) by 
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0)))

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(205) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), s(z0)) -> c7(F(g(h, s(x0)), s(s(g(d, z0)))), DOUBLE(s(z0))) by 
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))), DOUBLE(s(0)))
   F(s(0), s(x1)) -> c7(F(0, s(s(g(d, x1)))), DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(x0), 0) -> c7(F(g(h, s(x0)), 0))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(209) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) by 
   F(s(0), 0) -> c7(F(0, 0))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(0), 0) -> c7(F(0, 0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), 0) -> c7(F(0, 0))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(213) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(0))), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1)) by 
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(215) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), double(x1)), DOUBLE(x1)) by 
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(217) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(0)), x1) -> c7(F(s(0), double(x1)), DOUBLE(x1)) by 
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(219) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0))) by 
   F(s(0), s(0)) -> c6(F(0, s(s(0))), HALF(s(0)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(0), s(0)) -> c6(F(0, s(s(0))), HALF(s(0)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(0)) -> c6(F(0, s(s(0))), HALF(s(0)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(223) 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(0)) -> c6
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = 0
   POL(HALF(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = [3]
   POL(double(x_1)) = [1] + [2]x_1
   POL(g(x_1, x_2)) = [3] + [3]x_1 + [2]x_2
   POL(h) = 0
   POL(s(x_1)) = [2]

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(0)) -> c6
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(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(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(0)) -> c6
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(227) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0))) by 
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(0), s(s(x1))) -> c6(F(0, s(s(s(s(g(d, x1)))))), HALF(s(0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))

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

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(0)) -> c6
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(0), s(s(x1))) -> c6(F(0, s(s(s(s(g(d, x1)))))), HALF(s(0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(0), s(s(x1))) -> c6(F(0, s(s(s(s(g(d, x1)))))), HALF(s(0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(0)) -> c6
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(229) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(0)) -> c6

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

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(0), s(s(x1))) -> c6(F(0, s(s(s(s(g(d, x1)))))), HALF(s(0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(0), s(s(x1))) -> c6(F(0, s(s(s(s(g(d, x1)))))), HALF(s(0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(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(0), s(s(x1))) -> c6
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = x_1 + x_2
   POL(HALF(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = 0
   POL(s(x_1)) = 0

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

(234)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(235) 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(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6) = 0
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(236)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(237) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0)))) by 
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(0), s(s(x1))) -> c7(F(0, s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))

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

(238)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(0), s(s(x1))) -> c7(F(0, s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(0), s(s(x1))) -> c7(F(0, s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(0), s(s(x1))) -> c6
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6


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

(239) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(s(x1))) -> c6

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

(240)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(0), s(s(x1))) -> c7(F(0, s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(0), s(s(x1))) -> c7(F(0, s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(242)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(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(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = x_2
   POL(HALF(x_1)) = x_1
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = [1]
   POL(s(x_1)) = 0

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

(244)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(245) 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(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(246)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

(247) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0)))) by 
   F(s(0), s(0)) -> c7(F(0, s(s(0))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))

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

(248)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(0)) -> c7(F(0, s(s(0))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(0), s(0)) -> c7(F(0, s(s(0))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2


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

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

(250)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(251) 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(0)) -> c7
We considered the (Usable) Rules:none
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1]
   POL(G(x_1, x_2)) = x_2
   POL(HALF(x_1)) = x_1
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = 0
   POL(h) = [1]
   POL(s(x_1)) = 0

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

(252)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(0), s(0)) -> c7
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(253) 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(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
We considered the (Usable) Rules:
   g(z0, 0) -> 0
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
And the Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = x_1
   POL(HALF(x_1)) = 0
   POL(c1(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(c6(x_1, x_2)) = x_1 + x_2
   POL(c7) = 0
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(d) = 0
   POL(double(x_1)) = [1] + x_1
   POL(g(x_1, x_2)) = x_2
   POL(h) = 0
   POL(s(x_1)) = [1] + x_1

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

(254)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(x0), x1) -> c6(HALF(s(x0)))
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(0), s(0)) -> c7
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c6_1, c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7


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

(255) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(x0), x1) -> c6(HALF(s(x0))) by 
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))

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

(256)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(0), s(0)) -> c7
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(0), s(0)) -> c7
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c7, c6_1


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

(257) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(0)) -> c7

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

(258)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(x0), x1) -> c7(DOUBLE(x1))
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6_1


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

(259) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(x0), x1) -> c7(DOUBLE(x1)) by 
   F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0))))

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

(260)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6_1


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

(261) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) by 
   F(s(s(x0)), 0) -> c6(F(s(g(h, x0)), 0), HALF(s(s(x0))))
   F(s(s(x0)), s(z0)) -> c6(F(s(g(h, x0)), s(s(g(d, z0)))), HALF(s(s(x0))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(s(0)))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(0))))

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

(262)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(s(0)))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6_1


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

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

(264)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c6_2, c1_1, c3_1, c4_1, c5_1, c7_2, c6_1


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

(265) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), g(d, x1)), HALF(s(s(z0)))) by 
   F(s(s(x0)), 0) -> c6(F(s(g(h, x0)), 0), HALF(s(s(x0))))
   F(s(s(x0)), s(z0)) -> c6(F(s(g(h, x0)), s(s(g(d, z0)))), HALF(s(s(x0))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(s(0)))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(0))))

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

(266)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(s(0)))))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)), HALF(s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

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

(268)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

(269) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1)) by 
   F(s(s(x0)), 0) -> c7(F(s(g(h, x0)), 0), DOUBLE(0))
   F(s(s(x0)), s(z0)) -> c7(F(s(g(h, x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(s(0))), x1) -> c7(F(s(0), g(d, x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), g(d, x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), g(d, x1)), DOUBLE(x1))

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

(270)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(x0)), 0) -> c7(F(s(g(h, x0)), 0), DOUBLE(0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

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

(272)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c7_2, c6_2, c6_1


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

(273) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(x0)), z0) -> c7(F(s(g(h, x0)), g(d, z0)), DOUBLE(z0)) by 
   F(s(s(x0)), 0) -> c7(F(s(g(h, x0)), 0), DOUBLE(0))
   F(s(s(x0)), s(z0)) -> c7(F(s(g(h, x0)), s(s(g(d, z0)))), DOUBLE(s(z0)))
   F(s(s(s(0))), x1) -> c7(F(s(0), g(d, x1)), DOUBLE(x1))
   F(s(s(s(s(z0)))), x1) -> c7(F(s(s(g(h, z0))), g(d, x1)), DOUBLE(x1))
   F(s(s(0)), x1) -> c7(F(s(0), g(d, x1)), DOUBLE(x1))

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

(274)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(x0)), 0) -> c7(F(s(g(h, x0)), 0), DOUBLE(0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

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

(276)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(277) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) by 
   F(s(s(s(0))), 0) -> c6(F(s(0), 0), HALF(s(s(s(0)))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(0)), 0) -> c6(F(s(0), 0), HALF(s(s(0))))

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

(278)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0), HALF(s(s(s(0)))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(0)), 0) -> c6(F(s(0), 0), HALF(s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

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

(280)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(281) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) by 
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))

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

(282)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(283) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(x0)), 0) -> c6(F(s(g(h, x0)), 0), HALF(s(s(x0)))) by 
   F(s(s(s(0))), 0) -> c6(F(s(0), 0), HALF(s(s(s(0)))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(0)), 0) -> c6(F(s(0), 0), HALF(s(s(0))))

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

(284)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0), HALF(s(s(s(0)))))
   F(s(s(0)), 0) -> c6(F(s(0), 0), HALF(s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

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

(286)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(287) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0)))))) by 
   F(s(s(s(s(x0)))), 0) -> c6(F(s(s(g(h, x0))), 0), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(x0)))), x1) -> c6(HALF(s(s(s(s(x0))))))

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

(288)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(289) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(s(s(s(0))))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1)) by F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
----------------------------------------

(290)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(291) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0))) by 
   F(s(0), s(0)) -> c6(F(g(h, s(0)), s(s(0))), HALF(s(0)))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))), HALF(s(s(0))))

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

(292)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(0), s(0)) -> c6(F(g(h, s(0)), s(s(0))), HALF(s(0)))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))), HALF(s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

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

(294)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(0), s(0)) -> c6(F(g(h, s(0)), s(s(0))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(295) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(0)) -> c6(F(g(h, s(0)), s(s(0)))) by 
   F(s(0), s(0)) -> c6(F(0, s(s(0))))

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

(296)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))))
   F(s(0), s(0)) -> c6(F(0, s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0))))
   F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(297) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(0)) -> c6(F(0, s(s(0))))

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

(298)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(299) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0)))) by 
   F(s(0), s(0)) -> c7(F(g(h, s(0)), s(s(0))))
   F(s(s(y0)), s(0)) -> c7(F(g(h, s(s(y0))), s(s(0))))
   F(s(s(s(y0))), s(0)) -> c7(F(g(h, s(s(s(y0)))), s(s(0))))
   F(s(s(0)), s(0)) -> c7(F(g(h, s(s(0))), s(s(0))))

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

(300)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))))
   F(s(0), s(0)) -> c7(F(g(h, s(0)), s(s(0))))
   F(s(s(y0)), s(0)) -> c7(F(g(h, s(s(y0))), s(s(0))))
   F(s(s(s(y0))), s(0)) -> c7(F(g(h, s(s(s(y0)))), s(s(0))))
   F(s(s(0)), s(0)) -> c7(F(g(h, s(s(0))), s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(301) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(0)) -> c7(F(g(h, s(0)), s(s(0)))) by 
   F(s(0), s(0)) -> c7(F(0, s(s(0))))

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

(302)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))))
   F(s(s(y0)), s(0)) -> c7(F(g(h, s(s(y0))), s(s(0))))
   F(s(s(s(y0))), s(0)) -> c7(F(g(h, s(s(s(y0)))), s(s(0))))
   F(s(s(0)), s(0)) -> c7(F(g(h, s(s(0))), s(s(0))))
   F(s(0), s(0)) -> c7(F(0, s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

(303) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(0)) -> c7(F(0, s(s(0))))

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

(304)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   double(z0) -> g(d, z0)
Tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(x0), s(s(z0))) -> c6(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), HALF(s(x0)))
   F(s(x0), s(s(z0))) -> c7(F(g(h, s(x0)), s(s(s(s(g(d, z0)))))), DOUBLE(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(s(s(0))), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(x0)))), z0) -> c7(F(s(s(g(h, x0))), g(d, z0)), DOUBLE(z0))
   F(s(s(s(s(s(s(z0)))))), x1) -> c7(F(s(s(s(g(h, z0)))), double(x1)), DOUBLE(x1))
   F(s(s(s(s(0)))), x1) -> c7(F(s(s(0)), double(x1)), DOUBLE(x1))
   F(s(s(0)), z0) -> c7(F(s(0), g(d, z0)), DOUBLE(z0))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
   F(s(s(s(0))), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(0)), x1) -> c6(F(s(0), g(d, x1)))
   F(s(s(s(s(z0)))), x1) -> c6(F(s(s(g(h, z0))), g(d, x1)), HALF(s(s(s(s(z0))))))
   F(s(s(s(s(z0)))), 0) -> c6(F(s(s(g(h, z0))), 0), HALF(s(s(s(s(z0))))))
   F(s(s(s(0))), 0) -> c6(F(s(0), 0))
   F(s(s(0)), 0) -> c6(F(s(0), 0))
   F(s(s(s(0))), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(z0)))), 0) -> c7(F(s(s(g(h, z0))), 0))
   F(s(s(0)), 0) -> c7(F(s(0), 0))
   F(s(s(s(s(x0)))), s(z0)) -> c6(F(s(s(g(h, x0))), s(s(g(d, z0)))), HALF(s(s(s(s(x0))))))
   F(s(s(s(s(s(0))))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(s(0)))))))
   F(s(s(s(s(s(s(z0)))))), x1) -> c6(F(s(s(s(g(h, z0)))), g(d, x1)), HALF(s(s(s(s(s(s(z0))))))))
   F(s(s(s(s(0)))), x1) -> c6(F(s(s(0)), g(d, x1)), HALF(s(s(s(s(0))))))
   F(s(s(s(s(s(0))))), z0) -> c7(F(s(s(0)), g(d, z0)), DOUBLE(z0))
   F(s(s(y0)), s(0)) -> c6(F(g(h, s(s(y0))), s(s(0))), HALF(s(s(y0))))
   F(s(s(s(y0))), s(0)) -> c6(F(g(h, s(s(s(y0)))), s(s(0))), HALF(s(s(s(y0)))))
   F(s(s(0)), s(0)) -> c6(F(g(h, s(s(0))), s(s(0))))
   F(s(s(y0)), s(0)) -> c7(F(g(h, s(s(y0))), s(s(0))))
   F(s(s(s(y0))), s(0)) -> c7(F(g(h, s(s(s(y0)))), s(s(0))))
   F(s(s(0)), s(0)) -> c7(F(g(h, s(s(0))), s(s(0))))
S tuples:
   G(d, s(s(y0))) -> c1(G(d, s(y0)))
   G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0))))
   DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0))))
   HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0))))))
   F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0)))
   F(s(x0), s(s(s(z0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), HALF(s(x0)))
   F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0))))), DOUBLE(s(s(0))))
   F(s(x0), s(s(s(z0)))) -> c7(F(g(h, s(x0)), s(s(s(s(s(s(g(d, z0)))))))), DOUBLE(s(s(s(z0)))))
K tuples:
   F(s(0), x1) -> c7(DOUBLE(x1))
   F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0))))
   F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0))))
   F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0))
   F(s(0), s(x1)) -> c7(DOUBLE(s(x1)))
   F(s(s(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1)))
   F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0))))
   F(s(s(z0)), s(s(x1))) -> c6(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), HALF(s(s(z0))))
   F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(0), s(s(x1))) -> c7(DOUBLE(s(s(x1))))
   F(s(s(z0)), s(s(x1))) -> c7(F(s(g(h, z0)), s(s(s(s(g(d, x1)))))), DOUBLE(s(s(x1))))
   F(s(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0))))
   F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:   g_2, double_1

Defined Pair Symbols:   F_2, G_2, DOUBLE_1, HALF_1

Compound Symbols:   c7_1, c1_1, c3_1, c4_1, c5_1, c6_2, c7_2, c6_1


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

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

(306)
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:

   g(x, 0) -> 0 [1]
   g(d, s(x)) -> s(s(g(d, x))) [1]
   g(h, s(0)) -> 0 [1]
   g(h, s(s(x))) -> s(g(h, x)) [1]
   double(x) -> g(d, x) [1]
   half(x) -> g(h, x) [1]
   f(s(x), y) -> f(half(s(x)), double(y)) [1]
   f(s(0), y) -> y [1]
   id(x) -> f(x, s(0)) [1]

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

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

(308)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(x, 0) -> 0 [1]
   g(d, s(x)) -> s(s(g(d, x))) [1]
   g(h, s(0)) -> 0 [1]
   g(h, s(s(x))) -> s(g(h, x)) [1]
   double(x) -> g(d, x) [1]
   half(x) -> g(h, x) [1]
   f(s(x), y) -> f(half(s(x)), double(y)) [1]
   f(s(0), y) -> y [1]
   id(x) -> f(x, s(0)) [1]

The TRS has the following type information:
g :: d:h -> 0:s -> 0:s
0 :: 0:s
d :: d:h
s :: 0:s -> 0:s
h :: d:h
double :: 0:s -> 0:s
half :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s
id :: 0:s -> 0:s

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

(309) 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
   id_1

(c) The following functions are completely defined:

   half_1
   double_1
   g_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(310)
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:

   g(x, 0) -> 0 [1]
   g(d, s(x)) -> s(s(g(d, x))) [1]
   g(h, s(0)) -> 0 [1]
   g(h, s(s(x))) -> s(g(h, x)) [1]
   double(x) -> g(d, x) [1]
   half(x) -> g(h, x) [1]
   f(s(x), y) -> f(half(s(x)), double(y)) [1]
   f(s(0), y) -> y [1]
   id(x) -> f(x, s(0)) [1]

The TRS has the following type information:
g :: d:h -> 0:s -> 0:s
0 :: 0:s
d :: d:h
s :: 0:s -> 0:s
h :: d:h
double :: 0:s -> 0:s
half :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s
id :: 0:s -> 0:s

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

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

(312)
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:

   g(x, 0) -> 0 [1]
   g(d, s(x)) -> s(s(g(d, x))) [1]
   g(h, s(0)) -> 0 [1]
   g(h, s(s(x))) -> s(g(h, x)) [1]
   double(x) -> g(d, x) [1]
   half(x) -> g(h, x) [1]
   f(s(x), y) -> f(g(h, s(x)), g(d, y)) [3]
   f(s(0), y) -> y [1]
   id(x) -> f(x, s(0)) [1]

The TRS has the following type information:
g :: d:h -> 0:s -> 0:s
0 :: 0:s
d :: d:h
s :: 0:s -> 0:s
h :: d:h
double :: 0:s -> 0:s
half :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s
id :: 0:s -> 0:s

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

(313) 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
   d => 0
   h => 1

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

(314)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, x) :|: x >= 0, z = x
   f(z, z') -{ 1 }-> y :|: z = 1 + 0, y >= 0, z' = y
   f(z, z') -{ 3 }-> f(g(1, 1 + x), g(0, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   g(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 }-> 1 + g(1, x) :|: z = 1, x >= 0, z' = 1 + (1 + x)
   g(z, z') -{ 1 }-> 1 + (1 + g(0, x)) :|: z' = 1 + x, x >= 0, z = 0
   half(z) -{ 1 }-> g(1, x) :|: x >= 0, z = x
   id(z) -{ 1 }-> f(x, 1 + 0) :|: x >= 0, z = x


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

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

(316)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, z) :|: z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 3 }-> f(g(1, 1 + (z - 1)), g(0, z')) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 1 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 1 }-> g(1, z) :|: z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0


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

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

   { g }
   { double }
   { f }
   { half }
   { id }

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

(318)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, z) :|: z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 3 }-> f(g(1, 1 + (z - 1)), g(0, z')) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 1 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 1 }-> g(1, z) :|: z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {g}, {double}, {f}, {half}, {id}

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

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

(320)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, z) :|: z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 3 }-> f(g(1, 1 + (z - 1)), g(0, z')) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 1 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 1 }-> g(1, z) :|: z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {g}, {double}, {f}, {half}, {id}

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

(321) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(322)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, z) :|: z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 3 }-> f(g(1, 1 + (z - 1)), g(0, z')) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 1 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 1 }-> g(1, z) :|: z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {g}, {double}, {f}, {half}, {id}
Previous analysis results are:
  g: runtime: ?, size: O(n^1) [2*z']

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

(323) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(324)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, z) :|: z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 3 }-> f(g(1, 1 + (z - 1)), g(0, z')) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 }-> 1 + g(1, z' - 2) :|: z = 1, z' - 2 >= 0
   g(z, z') -{ 1 }-> 1 + (1 + g(0, z' - 1)) :|: z' - 1 >= 0, z = 0
   half(z) -{ 1 }-> g(1, z) :|: z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {double}, {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']

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

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

(326)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= 2 * z, z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 7 + z + z' }-> f(s2, s3) :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 2 * z, z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {double}, {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']

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

(327) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(328)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= 2 * z, z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 7 + z + z' }-> f(s2, s3) :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 2 * z, z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {double}, {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']
  double: runtime: ?, size: O(n^1) [2*z]

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

(329) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: double
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z

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

(330)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= 2 * z, z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 7 + z + z' }-> f(s2, s3) :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 2 * z, z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']
  double: runtime: O(n^1) [3 + z], size: O(n^1) [2*z]

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

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

(332)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= 2 * z, z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 7 + z + z' }-> f(s2, s3) :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 2 * z, z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']
  double: runtime: O(n^1) [3 + z], size: O(n^1) [2*z]

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

(333) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(334)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= 2 * z, z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 7 + z + z' }-> f(s2, s3) :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 2 * z, z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']
  double: runtime: O(n^1) [3 + z], size: O(n^1) [2*z]
  f: runtime: ?, size: INF

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

(335) 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: ?

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

(336)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= 2 * z, z >= 0
   f(z, z') -{ 1 }-> z' :|: z = 1 + 0, z' >= 0
   f(z, z') -{ 7 + z + z' }-> f(s2, s3) :|: s2 >= 0, s2 <= 2 * (1 + (z - 1)), s3 >= 0, s3 <= 2 * z', z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= 2 * (z' - 2), z = 1, z' - 2 >= 0
   g(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0, z = 0
   half(z) -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= 2 * z, z >= 0
   id(z) -{ 1 }-> f(z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {f}, {half}, {id}
Previous analysis results are:
  g: runtime: O(n^1) [2 + z'], size: O(n^1) [2*z']
  double: runtime: O(n^1) [3 + z], size: O(n^1) [2*z]
  f: runtime: INF, size: INF

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

(337) 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:

   f(v0, v1) -> null_f [0]
   g(v0, v1) -> null_g [0]

And the following fresh constants:    null_f, null_g

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

(338)
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:

   g(x, 0) -> 0 [1]
   g(d, s(x)) -> s(s(g(d, x))) [1]
   g(h, s(0)) -> 0 [1]
   g(h, s(s(x))) -> s(g(h, x)) [1]
   double(x) -> g(d, x) [1]
   half(x) -> g(h, x) [1]
   f(s(x), y) -> f(half(s(x)), double(y)) [1]
   f(s(0), y) -> y [1]
   id(x) -> f(x, s(0)) [1]
   f(v0, v1) -> null_f [0]
   g(v0, v1) -> null_g [0]

The TRS has the following type information:
g :: d:h -> 0:s:null_f:null_g -> 0:s:null_f:null_g
0 :: 0:s:null_f:null_g
d :: d:h
s :: 0:s:null_f:null_g -> 0:s:null_f:null_g
h :: d:h
double :: 0:s:null_f:null_g -> 0:s:null_f:null_g
half :: 0:s:null_f:null_g -> 0:s:null_f:null_g
f :: 0:s:null_f:null_g -> 0:s:null_f:null_g -> 0:s:null_f:null_g
id :: 0:s:null_f:null_g -> 0:s:null_f:null_g
null_f :: 0:s:null_f:null_g
null_g :: 0:s:null_f:null_g

Rewrite Strategy: INNERMOST
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(339) 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
   d => 0
   h => 1
   null_f => 0
   null_g => 0

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

(340)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> g(0, x) :|: x >= 0, z = x
   f(z, z') -{ 1 }-> y :|: z = 1 + 0, y >= 0, z' = y
   f(z, z') -{ 1 }-> f(half(1 + x), double(y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 1, z' = 1 + 0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 1 }-> 1 + g(1, x) :|: z = 1, x >= 0, z' = 1 + (1 + x)
   g(z, z') -{ 1 }-> 1 + (1 + g(0, x)) :|: z' = 1 + x, x >= 0, z = 0
   half(z) -{ 1 }-> g(1, x) :|: x >= 0, z = x
   id(z) -{ 1 }-> f(x, 1 + 0) :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.