KILLED proof of input_Eie3Mp9Ip3.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 426 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 195 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 3526 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 2447 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 12 ms] (58) CdtProblem (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 9 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 ms] (72) CdtProblem (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] (132) CdtProblem (133) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 ms] (154) CdtProblem (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (172) CdtProblem (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms] (200) CdtProblem (201) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms] (210) CdtProblem (211) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (212) CdtProblem (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (242) CdtProblem (243) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (244) CdtProblem (245) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (246) CdtProblem (247) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (248) CdtProblem (249) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (250) CdtProblem (251) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (252) CdtProblem (253) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (254) CdtProblem (255) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (256) CdtProblem (257) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (258) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) 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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) 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 ---------------------------------------- (9) 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 ---------------------------------------- (10) 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 ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) 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 ---------------------------------------- (13) 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 ---------------------------------------- (14) 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 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) 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 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { double } { f } { half } { id } ---------------------------------------- (18) 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} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) 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} ---------------------------------------- (21) 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' ---------------------------------------- (22) 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'] ---------------------------------------- (23) 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' ---------------------------------------- (24) 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'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) 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'] ---------------------------------------- (27) 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 ---------------------------------------- (28) 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] ---------------------------------------- (29) 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 ---------------------------------------- (30) 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] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) 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] ---------------------------------------- (33) 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: ? ---------------------------------------- (34) 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 ---------------------------------------- (35) 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: ? ---------------------------------------- (36) 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 ---------------------------------------- (37) 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 ---------------------------------------- (38) 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 ---------------------------------------- (39) 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 ---------------------------------------- (40) 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. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) 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 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ID(z0) -> c9(F(z0, s(0))) Removed 3 trailing nodes: G(h, s(0)) -> c2 G(z0, 0) -> c F(s(0), z0) -> c8 ---------------------------------------- (44) 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 ---------------------------------------- (45) 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)) ---------------------------------------- (46) 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 ---------------------------------------- (47) 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))) ---------------------------------------- (48) 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 ---------------------------------------- (49) 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)) ---------------------------------------- (50) 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 ---------------------------------------- (51) 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))) ---------------------------------------- (52) 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 ---------------------------------------- (53) 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))) ---------------------------------------- (54) 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 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) 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 ---------------------------------------- (57) 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 ---------------------------------------- (58) 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 ---------------------------------------- (59) 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 ---------------------------------------- (60) 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 ---------------------------------------- (61) 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)) ---------------------------------------- (62) 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 ---------------------------------------- (63) 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 ---------------------------------------- (64) 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 ---------------------------------------- (65) 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)) ---------------------------------------- (66) 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 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) 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 ---------------------------------------- (69) 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) = [1] 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)) = [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 ---------------------------------------- (70) 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 ---------------------------------------- (71) 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 ---------------------------------------- (72) 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 ---------------------------------------- (73) 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))) ---------------------------------------- (74) 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 ---------------------------------------- (75) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HALF(z0) -> c5(G(h, z0)) by HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) HALF(s(0)) -> c5(G(h, s(0))) ---------------------------------------- (76) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) HALF(s(0)) -> c5(G(h, 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) HALF(s(0)) -> c5(G(h, s(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, F_2, HALF_1 Compound Symbols: c1_1, c3_1, c4_1, c6_2, c6_1, c7_2, c7_1, c5_1 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: HALF(s(0)) -> c5(G(h, s(0))) ---------------------------------------- (78) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(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)))) 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, F_2, HALF_1 Compound Symbols: c1_1, c3_1, c4_1, c6_2, c6_1, c7_2, c7_1, c5_1 ---------------------------------------- (79) 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))) ---------------------------------------- (80) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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, F_2, HALF_1 Compound Symbols: c3_1, c4_1, c6_2, c6_1, c7_2, c7_1, c5_1, c1_1 ---------------------------------------- (81) 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))) ---------------------------------------- (82) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) S tuples: G(h, s(s(z0))) -> c3(G(h, z0)) DOUBLE(z0) -> c4(G(d, 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) 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, F_2, HALF_1 Compound Symbols: c3_1, c4_1, c6_2, c6_1, c7_2, c7_1, c5_1, c1_1 ---------------------------------------- (83) 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)))) ---------------------------------------- (84) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) S tuples: DOUBLE(z0) -> c4(G(d, 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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, F_2, HALF_1, G_2 Compound Symbols: c4_1, c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1 ---------------------------------------- (85) 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)))) ---------------------------------------- (86) 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)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) S tuples: DOUBLE(z0) -> c4(G(d, 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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, F_2, HALF_1, G_2 Compound Symbols: c4_1, c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1 ---------------------------------------- (87) 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)))) ---------------------------------------- (88) 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), 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) S tuples: 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (90) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 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: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (91) 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))) ---------------------------------------- (92) 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), 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (93) 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)))) ---------------------------------------- (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: 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), 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)) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: half_1, g_2, double_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), x1) -> c7(DOUBLE(x1)) by F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) ---------------------------------------- (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: 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), 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(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: half_1, g_2, double_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) by F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) ---------------------------------------- (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: 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), 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: half_1, g_2, double_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (99) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: double(z0) -> g(d, z0) ---------------------------------------- (100) 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))) half(z0) -> g(h, 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), 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (101) 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)))) F(s(x0), x1) -> c6(HALF(s(x0))) ---------------------------------------- (102) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)))) F(s(x0), x1) -> c6(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)))) F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c7_2, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (104) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)))) F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c7_2, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (105) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(0), x1) -> c6(HALF(s(0))) ---------------------------------------- (106) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c7_2, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (107) 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)) half(z0) -> g(h, z0) And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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]x_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(g(x_1, x_2)) = x_2 POL(h) = 0 POL(half(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (108) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) 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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c7_2, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1 ---------------------------------------- (109) 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)) ---------------------------------------- (110) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(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)) S tuples: F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (112) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) F(s(0), x1) -> c7(DOUBLE(x1)) S tuples: F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (113) 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: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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)) = 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(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = 0 POL(h) = 0 POL(half(x_1)) = [1] + x_1 POL(s(x_1)) = 0 ---------------------------------------- (114) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) F(s(0), x1) -> c7(DOUBLE(x1)) S tuples: F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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(x0), 0) -> c7(F(g(h, s(x0)), 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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (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(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) We considered the (Usable) Rules: g(z0, 0) -> 0 g(h, s(0)) -> 0 g(h, s(s(z0))) -> s(g(h, z0)) half(z0) -> g(h, z0) And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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)) = 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(g(x_1, x_2)) = x_2 POL(h) = 0 POL(half(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (116) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) F(s(0), x1) -> c7(DOUBLE(x1)) S tuples: F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (117) 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)))) F(s(x0), 0) -> c6(HALF(s(x0))) ---------------------------------------- (118) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) F(s(0), x1) -> c7(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)))) F(s(x0), 0) -> c6(HALF(s(x0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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)))) F(s(x0), 0) -> c6(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)))) F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (120) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (121) 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), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) We considered the (Usable) Rules:none And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> 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)) = 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(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = 0 POL(h) = 0 POL(half(x_1)) = [1] + x_1 POL(s(x_1)) = 0 ---------------------------------------- (122) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (123) 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)) half(z0) -> g(h, z0) And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> 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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = x_2 POL(h) = 0 POL(half(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (124) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (125) 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)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) ---------------------------------------- (126) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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)))) F(s(x0), s(x1)) -> c6(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)))) F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (128) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (129) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(x0), s(0)) -> c6(F(g(h, s(x0)), s(s(0))), HALF(s(x0))) F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(s(z0)), x1) -> c6(F(s(g(h, z0)), g(d, x1)), HALF(s(s(z0)))) F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(s(z0)), x1) -> c7(F(s(g(h, z0)), g(d, x1)), DOUBLE(x1)) F(s(0), x1) -> c7(DOUBLE(x1)) ---------------------------------------- (130) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (131) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) We considered the (Usable) Rules:none And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(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)) = 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(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = 0 POL(h) = 0 POL(half(x_1)) = [1] + x_1 POL(s(x_1)) = 0 ---------------------------------------- (132) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) Defined Rule Symbols: g_2, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (133) 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)) half(z0) -> g(h, z0) And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = x_2 POL(h) = 0 POL(half(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (134) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(x0), 0) -> c7(F(half(s(x0)), 0)) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(half(s(x0)), 0)) 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_1, c7_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), 0) -> c7(F(half(s(x0)), 0)) by F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) ---------------------------------------- (136) 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))) half(z0) -> g(h, z0) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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, half_1 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_2, c7_1 ---------------------------------------- (137) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: half(z0) -> g(h, z0) ---------------------------------------- (138) 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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), 0) -> c6(F(g(h, s(z0)), 0), HALF(s(z0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c6_2, c5_1, c1_1, c3_1, c4_1, c7_2, c7_1 ---------------------------------------- (139) 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)))) F(s(x0), 0) -> c6(HALF(s(x0))) ---------------------------------------- (140) 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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(0), 0) -> c6(F(0, 0), HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c6_2, c3_1, c4_1, c7_2, c7_1 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) F(s(z0), s(z1)) -> c6(F(g(h, s(z0)), s(s(g(d, z1)))), HALF(s(z0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c6_2, c3_1, c4_1, c7_2, c7_1 ---------------------------------------- (143) 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)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(0), s(x1)) -> c6(F(0, s(s(g(d, x1)))), HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_2, c7_1 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(z1)) -> c7(F(g(h, s(z0)), s(s(g(d, z1)))), DOUBLE(s(z1))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), 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)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_2, c7_1 ---------------------------------------- (147) 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))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (149) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (151) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0)))) F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(0), s(x1)) -> c7(DOUBLE(s(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (153) 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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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)) = [1] POL(G(x_1, x_2)) = x_1 + x_2 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(g(x_1, x_2)) = 0 POL(h) = 0 POL(s(x_1)) = 0 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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))) 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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c7(F(g(h, s(x0)), s(s(0)))) F(s(0), s(x1)) -> c7(DOUBLE(s(x1))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (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)), 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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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)) = [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(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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(x0), 0) -> c7(F(g(h, s(x0)), 0)) 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)))) 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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (157) 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)) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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)) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(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(s(z0)), x1) -> c7(F(s(g(h, z0)), double(x1)), DOUBLE(x1)) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), 0) -> c7(F(0, 0)) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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)) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (161) 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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = x_2 POL(h) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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)))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (163) 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)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 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)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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)))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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)))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (167) 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)))) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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)))) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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)))) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (171) 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))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) We considered the (Usable) Rules:none And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(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)) = x_1 + x_2 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(g(x_1, x_2)) = 0 POL(h) = 0 POL(s(x_1)) = 0 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (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(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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(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] + 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(g(x_1, x_2)) = x_2 POL(h) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (175) 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)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) F(s(0), s(0)) -> c6(F(0, s(s(0))), HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (179) 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)))) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) F(s(0), s(s(x1))) -> c6(F(0, s(s(s(s(g(d, x1)))))), HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(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(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(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (183) 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)))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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)))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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)))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (187) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(x0), s(s(x1))) -> c7(DOUBLE(s(s(x1)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(x1))) -> c7(DOUBLE(s(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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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)))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (189) 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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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)))) 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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = x_2 POL(h) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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)))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (191) 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)))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(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: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) 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(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) 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(z0)), s(x1)) -> c7(F(s(g(h, z0)), s(s(g(d, x1)))), DOUBLE(s(x1))) F(s(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (193) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c7(F(0, s(s(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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(s(0))) -> c6(F(g(h, s(x0)), s(s(s(s(0))))), HALF(s(x0))) by F(s(0), s(s(0))) -> c6(F(0, s(s(s(s(0))))), HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(0), s(s(0))) -> c6(F(0, s(s(s(s(0))))), HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0))) -> c6(F(0, s(s(s(s(0))))), HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (199) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) We considered the (Usable) Rules:none And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> 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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = [3] POL(g(x_1, x_2)) = [3] + [3]x_1 + [2]x_2 POL(h) = 0 POL(s(x_1)) = [2] ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (201) 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(0))) -> c6(F(s(g(h, z0)), s(s(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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> 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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = x_2 POL(h) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 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))) by F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(F(0, s(s(s(s(s(s(g(d, x1)))))))), HALF(s(0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(F(0, s(s(s(s(s(s(g(d, x1)))))))), HALF(s(0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(F(0, s(s(s(s(s(s(g(d, x1)))))))), HALF(s(0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (205) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (207) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(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))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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)))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (209) 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(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) We considered the (Usable) Rules:none And the Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(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)) = x_1 + x_2 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(g(x_1, x_2)) = 0 POL(h) = 0 POL(s(x_1)) = 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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (211) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(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(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(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_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(d) = 0 POL(g(x_1, x_2)) = x_2 POL(h) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) S tuples: HALF(s(x0)) -> c5(G(h, s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) K tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HALF(s(x0)) -> c5(G(h, s(x0))) by HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) F(s(0), x1) -> c6(HALF(s(0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), s(x1)) -> c6(F(s(g(h, z0)), s(s(g(d, x1)))), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(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(s(z0)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(HALF(s(x0))) F(s(0), s(0)) -> c6(HALF(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(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(HALF(s(0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) 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(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(HALF(s(x0))) F(s(0), s(x1)) -> c6(HALF(s(0))) 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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(HALF(s(x0))) F(s(0), s(s(x1))) -> c6(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)))) 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(0))) -> c6(HALF(s(x0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (215) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: F(s(0), s(s(x1))) -> c6(HALF(s(0))) F(s(0), x1) -> c6(HALF(s(0))) F(s(0), s(s(s(x1)))) -> c6(HALF(s(0))) F(s(0), 0) -> c6(HALF(s(0))) F(s(0), s(x1)) -> c6(HALF(s(0))) F(s(0), s(s(0))) -> c6(HALF(s(0))) F(s(0), s(0)) -> c6(HALF(s(0))) ---------------------------------------- (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))) Tuples: F(s(x0), x1) -> c6(HALF(s(x0))) HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) 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(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_1, c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2 ---------------------------------------- (217) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(x0), x1) -> c6(HALF(s(x0))) by F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) G(d, s(s(y0))) -> c1(G(d, s(y0))) G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, F_2, DOUBLE_1 Compound Symbols: c5_1, c1_1, c3_1, c6_2, c4_1, c7_1, c7_2, c6_1 ---------------------------------------- (219) 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)))) F(s(s(x0)), x1) -> c6(HALF(s(s(x0)))) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c7_2, c6_2, c6_1 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) 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)) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) F(s(s(x0)), 0) -> c7(F(s(g(h, x0)), 0), DOUBLE(0)) 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)) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2 ---------------------------------------- (223) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) 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)) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), s(x1)) -> c6(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)) -> 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2 ---------------------------------------- (225) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(s(x0), s(x1)) -> c6(HALF(s(x0))) by F(s(y0), s(s(y1))) -> c6(HALF(s(y0))) F(s(y0), s(s(0))) -> c6(HALF(s(y0))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(0), s(z1)) -> c6(HALF(s(0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(0), s(z1)) -> c6(HALF(s(0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(s(z0)), z1) -> c6(F(s(g(h, z0)), g(d, z1)), HALF(s(s(z0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(s(z0)), z1) -> c7(F(s(g(h, z0)), g(d, z1)), DOUBLE(z1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(0), s(z1)) -> c6(HALF(s(0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2 ---------------------------------------- (227) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(z1)) -> c6(HALF(s(0))) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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(x0), s(x1)) -> c6(HALF(s(x0))) 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2 ---------------------------------------- (229) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(s(x0), s(x1)) -> c6(HALF(s(x0))) by F(s(y0), s(s(y1))) -> c6(HALF(s(y0))) F(s(y0), s(s(0))) -> c6(HALF(s(y0))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(0), s(z1)) -> c6(HALF(s(0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) F(s(0), s(z1)) -> c6(HALF(s(0))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2 ---------------------------------------- (231) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(z1)) -> c6(HALF(s(0))) ---------------------------------------- (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))) Tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) S tuples: HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: HALF_1, G_2, DOUBLE_1, F_2 Compound Symbols: c5_1, c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2 ---------------------------------------- (233) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HALF(s(s(x0))) -> c5(G(h, s(s(x0)))) by HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) ---------------------------------------- (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))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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)))) 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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)))) 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: G_2, DOUBLE_1, F_2, HALF_1 Compound Symbols: c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2, c5_1 ---------------------------------------- (235) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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))) 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: G_2, DOUBLE_1, F_2, HALF_1 Compound Symbols: c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2, c5_1 ---------------------------------------- (237) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(s(x0), s(0)) -> c6(HALF(s(x0))) by F(s(s(y0)), s(0)) -> c6(HALF(s(s(y0)))) F(s(0), s(0)) -> c6(HALF(s(0))) ---------------------------------------- (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))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) F(s(s(z0)), s(0)) -> c6(F(s(g(h, z0)), s(s(0))), HALF(s(s(z0)))) F(s(x0), s(0)) -> c6(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(x1))) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) F(s(0), s(0)) -> c6(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)))) 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: G_2, DOUBLE_1, F_2, HALF_1 Compound Symbols: c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2, c5_1 ---------------------------------------- (239) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c6(HALF(s(0))) ---------------------------------------- (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))) 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(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)))) 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: G_2, DOUBLE_1, F_2, HALF_1 Compound Symbols: c1_1, c3_1, c4_1, c7_1, c6_2, c6_1, c7_2, c5_1 ---------------------------------------- (241) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(d, s(s(y0))) -> c1(G(d, s(y0))) by G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) ---------------------------------------- (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))) Tuples: 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), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) S tuples: 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(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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: G_2, DOUBLE_1, F_2, HALF_1 Compound Symbols: c3_1, c4_1, c7_1, c6_2, c6_1, c7_2, c5_1, c1_1 ---------------------------------------- (243) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(h, s(s(s(s(y0))))) -> c3(G(h, s(s(y0)))) by G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) ---------------------------------------- (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))) Tuples: DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) S tuples: DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: DOUBLE_1, F_2, HALF_1, G_2 Compound Symbols: c4_1, c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1 ---------------------------------------- (245) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(s(y0))) -> c4(G(d, s(s(y0)))) by DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) ---------------------------------------- (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))) Tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (247) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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))) Tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) K tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (249) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) by F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) ---------------------------------------- (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))) Tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) K tuples: F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (251) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) by F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) ---------------------------------------- (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))) Tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) F(s(x0), s(0)) -> c6(HALF(s(x0))) 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(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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) K tuples: F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (253) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(s(x0), s(0)) -> c6(HALF(s(x0))) by F(s(s(y0)), s(0)) -> c6(HALF(s(s(y0)))) F(s(0), s(0)) -> c6(HALF(s(0))) ---------------------------------------- (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))) Tuples: F(s(z0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) F(s(0), s(s(y0))) -> c7(DOUBLE(s(s(y0)))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) 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(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(s(z0)), s(0)) -> c7(F(s(g(h, z0)), s(s(0)))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(0)) -> c6(HALF(s(0))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) K tuples: F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (255) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0), s(0)) -> c6(HALF(s(0))) ---------------------------------------- (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))) 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(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) 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(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(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) K tuples: F(s(0), x1) -> c7(DOUBLE(x1)) F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c7_1, c6_2, c6_1, c7_2, c5_1, c1_1, c3_1, c4_1 ---------------------------------------- (257) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(0), x1) -> c7(DOUBLE(x1)) by F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) ---------------------------------------- (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))) Tuples: F(s(s(z0)), 0) -> c6(F(s(g(h, z0)), 0), HALF(s(s(z0)))) F(s(x0), 0) -> c6(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)), 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) -> c7(F(s(g(h, z0)), 0)) 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))) -> c6(HALF(s(x0))) 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(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(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(0))) -> c6(HALF(s(x0))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) 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) -> 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)) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(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(y0)), s(0)) -> c6(HALF(s(s(y0)))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) S tuples: 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(x0), s(s(s(s(z0))))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(s(s(g(d, z0)))))))))), HALF(s(x0))) HALF(s(s(s(s(y0))))) -> c5(G(h, s(s(s(s(y0)))))) G(d, s(s(s(y0)))) -> c1(G(d, s(s(y0)))) G(h, s(s(s(s(s(s(y0))))))) -> c3(G(h, s(s(s(s(y0)))))) DOUBLE(s(s(s(y0)))) -> c4(G(d, s(s(s(y0))))) F(s(x0), s(s(0))) -> c7(F(g(h, s(x0)), s(s(s(s(0)))))) K tuples: F(s(x0), 0) -> c6(HALF(s(x0))) 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), 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)), 0) -> c7(F(s(g(h, z0)), 0)) F(s(x0), s(s(x1))) -> c6(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(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(0))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(0))) -> c6(F(s(g(h, z0)), s(s(s(s(0))))), HALF(s(s(z0)))) F(s(x0), s(s(s(0)))) -> c6(F(g(h, s(x0)), s(s(s(s(s(s(0))))))), HALF(s(x0))) F(s(x0), s(s(s(x1)))) -> c6(HALF(s(x0))) F(s(s(z0)), s(s(s(x1)))) -> c6(F(s(g(h, z0)), s(s(s(s(s(s(g(d, x1)))))))), HALF(s(s(z0)))) F(s(s(y0)), z1) -> c6(HALF(s(s(y0)))) F(s(s(s(s(y0)))), z1) -> c6(HALF(s(s(s(s(y0)))))) F(s(y0), s(s(s(s(y1))))) -> c6(HALF(s(y0))) F(s(z0), s(s(s(s(0))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(y1))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(0))))))) -> c6(HALF(s(z0))) F(s(z0), s(s(s(s(s(s(s(s(y1))))))))) -> c6(HALF(s(z0))) F(s(s(y0)), s(z1)) -> c6(HALF(s(s(y0)))) F(s(z0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) F(s(0), s(s(s(y0)))) -> c7(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: g_2 Defined Pair Symbols: F_2, HALF_1, G_2, DOUBLE_1 Compound Symbols: c6_2, c6_1, c7_2, c7_1, c5_1, c1_1, c3_1, c4_1