KILLED proof of input_ThNuQyJdtt.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) InliningProof [UPPER BOUND(ID), 750 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 166 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 88 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 121 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 152 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 580 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 113 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 3250 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 6592 ms] (52) CpxRNTS (53) CompletionProof [UPPER BOUND(ID), 0 ms] (54) CpxTypedWeightedCompleteTrs (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (66) CdtProblem (67) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 165 ms] (72) CdtProblem (73) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (92) CdtProblem (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 11 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 33 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 38 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 62 ms] (160) CdtProblem (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 38 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 22 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 48 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 63 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 66 ms] (172) CdtProblem (173) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (174) 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: f(0) -> 0 f(s(0)) -> s(0) f(s(s(x))) -> p(h(g(x))) g(0) -> pair(s(0), s(0)) g(s(x)) -> h(g(x)) h(x) -> pair(+(p(x), q(x)), p(x)) p(pair(x, y)) -> x q(pair(x, y)) -> y +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) f(s(s(x))) -> +(p(g(x)), q(g(x))) g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) 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: f(0') -> 0' f(s(0')) -> s(0') f(s(s(x))) -> p(h(g(x))) g(0') -> pair(s(0'), s(0')) g(s(x)) -> h(g(x)) h(x) -> pair(+'(p(x), q(x)), p(x)) p(pair(x, y)) -> x q(pair(x, y)) -> y +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) f(s(s(x))) -> +'(p(g(x)), q(g(x))) g(s(x)) -> pair(+'(p(g(x)), q(g(x))), p(g(x))) 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: f(0) -> 0 f(s(0)) -> s(0) f(s(s(x))) -> p(h(g(x))) g(0) -> pair(s(0), s(0)) g(s(x)) -> h(g(x)) h(x) -> pair(+(p(x), q(x)), p(x)) p(pair(x, y)) -> x q(pair(x, y)) -> y +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) f(s(s(x))) -> +(p(g(x)), q(g(x))) g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) 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: f(0) -> 0 [1] f(s(0)) -> s(0) [1] f(s(s(x))) -> p(h(g(x))) [1] g(0) -> pair(s(0), s(0)) [1] g(s(x)) -> h(g(x)) [1] h(x) -> pair(+(p(x), q(x)), p(x)) [1] p(pair(x, y)) -> x [1] q(pair(x, y)) -> y [1] +(x, 0) -> x [1] +(x, s(y)) -> s(+(x, y)) [1] f(s(s(x))) -> +(p(g(x)), q(g(x))) [1] g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (8) 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: f(0) -> 0 [1] f(s(0)) -> s(0) [1] f(s(s(x))) -> p(h(g(x))) [1] g(0) -> pair(s(0), s(0)) [1] g(s(x)) -> h(g(x)) [1] h(x) -> pair(plus(p(x), q(x)), p(x)) [1] p(pair(x, y)) -> x [1] q(pair(x, y)) -> y [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1] g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(0) -> 0 [1] f(s(0)) -> s(0) [1] f(s(s(x))) -> p(h(g(x))) [1] g(0) -> pair(s(0), s(0)) [1] g(s(x)) -> h(g(x)) [1] h(x) -> pair(plus(p(x), q(x)), p(x)) [1] p(pair(x, y)) -> x [1] q(pair(x, y)) -> y [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1] g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1] The TRS has the following type information: f :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: pair -> 0:s h :: pair -> pair g :: 0:s -> pair pair :: 0:s -> 0:s -> pair plus :: 0:s -> 0:s -> 0:s q :: pair -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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_1 (c) The following functions are completely defined: h_1 g_1 p_1 q_1 plus_2 Due to the following rules being added: p(v0) -> 0 [0] q(v0) -> 0 [0] And the following fresh constants: const ---------------------------------------- (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: f(0) -> 0 [1] f(s(0)) -> s(0) [1] f(s(s(x))) -> p(h(g(x))) [1] g(0) -> pair(s(0), s(0)) [1] g(s(x)) -> h(g(x)) [1] h(x) -> pair(plus(p(x), q(x)), p(x)) [1] p(pair(x, y)) -> x [1] q(pair(x, y)) -> y [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1] g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1] p(v0) -> 0 [0] q(v0) -> 0 [0] The TRS has the following type information: f :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: pair -> 0:s h :: pair -> pair g :: 0:s -> pair pair :: 0:s -> 0:s -> pair plus :: 0:s -> 0:s -> 0:s q :: pair -> 0:s const :: pair Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) 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: f(0) -> 0 [1] f(s(0)) -> s(0) [1] f(s(s(0))) -> p(h(pair(s(0), s(0)))) [2] f(s(s(s(x')))) -> p(h(h(g(x')))) [2] f(s(s(s(x'')))) -> p(h(pair(plus(p(g(x'')), q(g(x''))), p(g(x''))))) [2] g(0) -> pair(s(0), s(0)) [1] g(s(0)) -> h(pair(s(0), s(0))) [2] g(s(s(x1))) -> h(h(g(x1))) [2] g(s(s(x2))) -> h(pair(plus(p(g(x2)), q(g(x2))), p(g(x2)))) [2] h(pair(x3, y')) -> pair(plus(x3, y'), p(pair(x3, y'))) [3] h(pair(x3, y')) -> pair(plus(x3, 0), p(pair(x3, y'))) [2] h(pair(x4, y'')) -> pair(plus(0, y''), p(pair(x4, y''))) [2] h(x) -> pair(plus(0, 0), p(x)) [1] p(pair(x, y)) -> x [1] q(pair(x, y)) -> y [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] f(s(s(0))) -> plus(p(pair(s(0), s(0))), q(pair(s(0), s(0)))) [3] f(s(s(s(x5)))) -> plus(p(h(g(x5))), q(h(g(x5)))) [3] f(s(s(s(x5)))) -> plus(p(h(g(x5))), q(pair(plus(p(g(x5)), q(g(x5))), p(g(x5))))) [3] f(s(s(s(x6)))) -> plus(p(pair(plus(p(g(x6)), q(g(x6))), p(g(x6)))), q(h(g(x6)))) [3] f(s(s(s(x6)))) -> plus(p(pair(plus(p(g(x6)), q(g(x6))), p(g(x6)))), q(pair(plus(p(g(x6)), q(g(x6))), p(g(x6))))) [3] g(s(0)) -> pair(plus(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))) [4] g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(h(g(x7)))), p(h(g(x7)))) [4] g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(h(g(x7)))), p(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))) [4] g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))), p(h(g(x7)))) [4] g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))), p(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))) [4] g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(h(g(x8)))), p(h(g(x8)))) [4] g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(h(g(x8)))), p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))) [4] g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))), p(h(g(x8)))) [4] g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))), p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))) [4] p(v0) -> 0 [0] q(v0) -> 0 [0] The TRS has the following type information: f :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: pair -> 0:s h :: pair -> pair g :: 0:s -> pair pair :: 0:s -> 0:s -> pair plus :: 0:s -> 0:s -> 0:s q :: pair -> 0:s const :: pair Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 3 }-> plus(p(h(g(x5))), q(h(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0 f(z) -{ 3 }-> plus(p(h(g(x5))), q(1 + plus(p(g(x5)), q(g(x5))) + p(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(h(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6)) f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6)) f(z) -{ 3 }-> plus(p(1 + (1 + 0) + (1 + 0)), q(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 2 }-> p(h(h(g(x')))) :|: z = 1 + (1 + (1 + x')), x' >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(x'')), q(g(x''))) + p(g(x'')))) :|: x'' >= 0, z = 1 + (1 + (1 + x'')) f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(x1))) :|: z = 1 + (1 + x1), x1 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(x2)), q(g(x2))) + p(g(x2))) :|: z = 1 + (1 + x2), x2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + (1 + 0) + (1 + 0)), q(1 + (1 + 0) + (1 + 0))) + p(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 3 }-> 1 + plus(x3, y') + p(1 + x3 + y') :|: y' >= 0, z = 1 + x3 + y', x3 >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + p(1 + x3 + y') :|: y' >= 0, z = 1 + x3 + y', x3 >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + p(1 + x4 + y'') :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + p(x) :|: x >= 0, z = x p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (17) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 3 }-> plus(p(h(g(x5))), q(h(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0 f(z) -{ 3 }-> plus(p(h(g(x5))), q(1 + plus(p(g(x5)), q(g(x5))) + p(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(h(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6)) f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6)) f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 2 }-> p(h(h(g(x')))) :|: z = 1 + (1 + (1 + x')), x' >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(x'')), q(g(x''))) + p(g(x'')))) :|: x'' >= 0, z = 1 + (1 + (1 + x'')) f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(x1))) :|: z = 1 + (1 + x1), x1 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(x2)), q(g(x2))) + p(g(x2))) :|: z = 1 + (1 + x2), x2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8) g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: x >= 0, z = x, x = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: x >= 0, z = x, v0 >= 0, x = v0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { q } { p } { h } { g } { f } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {plus}, {q}, {p}, {h}, {g}, {f} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {plus}, {q}, {p}, {h}, {g}, {f} ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {plus}, {q}, {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {q}, {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {q}, {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: q after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {q}, {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: ?, size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: q after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {p}, {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {h}, {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] h: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + z ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 10 }-> s21 :|: s20 >= 0, s20 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), s21 >= 0, s21 <= s20, z = 1 + (1 + 0) f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 9 }-> s19 :|: s19 >= 0, s19 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: g after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 10 }-> s21 :|: s20 >= 0, s20 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), s21 >= 0, s21 <= s20, z = 1 + (1 + 0) f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 9 }-> s19 :|: s19 >= 0, s19 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z] g: runtime: ?, size: EXP ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 f(z) -{ 10 }-> s21 :|: s20 >= 0, s20 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), s21 >= 0, s21 <= s20, z = 1 + (1 + 0) f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0 f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 9 }-> s19 :|: s19 >= 0, s19 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + 0 g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0 g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0 g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0' g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0 g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0'' g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0 g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0 h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0 h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0 h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0 h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0 p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: z >= 0 Function symbols to be analyzed: {g}, {f} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] q: runtime: O(1) [1], size: O(n^1) [z] p: runtime: O(1) [1], size: O(n^1) [z] h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z] g: runtime: INF, size: EXP ---------------------------------------- (53) 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: p(v0) -> null_p [0] q(v0) -> null_q [0] f(v0) -> null_f [0] g(v0) -> null_g [0] plus(v0, v1) -> null_plus [0] And the following fresh constants: null_p, null_q, null_f, null_g, null_plus ---------------------------------------- (54) 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: f(0) -> 0 [1] f(s(0)) -> s(0) [1] f(s(s(x))) -> p(h(g(x))) [1] g(0) -> pair(s(0), s(0)) [1] g(s(x)) -> h(g(x)) [1] h(x) -> pair(plus(p(x), q(x)), p(x)) [1] p(pair(x, y)) -> x [1] q(pair(x, y)) -> y [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1] g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1] p(v0) -> null_p [0] q(v0) -> null_q [0] f(v0) -> null_f [0] g(v0) -> null_g [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: f :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus 0 :: 0:s:null_p:null_q:null_f:null_plus s :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus p :: pair:null_g -> 0:s:null_p:null_q:null_f:null_plus h :: pair:null_g -> pair:null_g g :: 0:s:null_p:null_q:null_f:null_plus -> pair:null_g pair :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus -> pair:null_g plus :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus q :: pair:null_g -> 0:s:null_p:null_q:null_f:null_plus null_p :: 0:s:null_p:null_q:null_f:null_plus null_q :: 0:s:null_p:null_q:null_f:null_plus null_f :: 0:s:null_p:null_q:null_f:null_plus null_g :: pair:null_g null_plus :: 0:s:null_p:null_q:null_f:null_plus Rewrite Strategy: INNERMOST ---------------------------------------- (55) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_p => 0 null_q => 0 null_f => 0 null_g => 0 null_plus => 0 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> plus(p(g(x)), q(g(x))) :|: x >= 0, z = 1 + (1 + x) f(z) -{ 1 }-> p(h(g(x))) :|: x >= 0, z = 1 + (1 + x) f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0 g(z) -{ 1 }-> h(g(x)) :|: x >= 0, z = 1 + x g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 g(z) -{ 1 }-> 1 + plus(p(g(x)), q(g(x))) + p(g(x)) :|: x >= 0, z = 1 + x g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0 h(z) -{ 1 }-> 1 + plus(p(x), q(x)) + p(x) :|: x >= 0, z = x p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (57) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: F(0) -> c F(s(0)) -> c1 F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0)) F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(0) -> c5 G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(s(z0)) -> c9(P(g(z0)), G(z0)) H(z0) -> c10(+'(p(z0), q(z0)), P(z0)) H(z0) -> c11(+'(p(z0), q(z0)), Q(z0)) H(z0) -> c12(P(z0)) P(pair(z0, z1)) -> c13 Q(pair(z0, z1)) -> c14 +'(z0, 0) -> c15 +'(z0, s(z1)) -> c16(+'(z0, z1)) S tuples: F(0) -> c F(s(0)) -> c1 F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0)) F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(0) -> c5 G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(s(z0)) -> c9(P(g(z0)), G(z0)) H(z0) -> c10(+'(p(z0), q(z0)), P(z0)) H(z0) -> c11(+'(p(z0), q(z0)), Q(z0)) H(z0) -> c12(P(z0)) P(pair(z0, z1)) -> c13 Q(pair(z0, z1)) -> c14 +'(z0, 0) -> c15 +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples:none Defined Rule Symbols: f_1, g_1, h_1, p_1, q_1, +_2 Defined Pair Symbols: F_1, G_1, H_1, P_1, Q_1, +'_2 Compound Symbols: c, c1, c2_3, c3_3, c4_3, c5, c6_2, c7_3, c8_3, c9_2, c10_2, c11_2, c12_1, c13, c14, c15, c16_1 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: H(z0) -> c12(P(z0)) +'(z0, 0) -> c15 F(s(0)) -> c1 Q(pair(z0, z1)) -> c14 F(0) -> c P(pair(z0, z1)) -> c13 G(0) -> c5 ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0)) F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(s(z0)) -> c9(P(g(z0)), G(z0)) H(z0) -> c10(+'(p(z0), q(z0)), P(z0)) H(z0) -> c11(+'(p(z0), q(z0)), Q(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) S tuples: F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0)) F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0)) G(s(z0)) -> c9(P(g(z0)), G(z0)) H(z0) -> c10(+'(p(z0), q(z0)), P(z0)) H(z0) -> c11(+'(p(z0), q(z0)), Q(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples:none Defined Rule Symbols: f_1, g_1, h_1, p_1, q_1, +_2 Defined Pair Symbols: F_1, G_1, H_1, +'_2 Compound Symbols: c2_3, c3_3, c4_3, c6_2, c7_3, c8_3, c9_2, c10_2, c11_2, c16_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) F(s(s(z0))) -> c2(H(g(z0)), G(z0)) F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), G(z0)) F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) S tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) F(s(s(z0))) -> c2(H(g(z0)), G(z0)) F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), G(z0)) F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) K tuples:none Defined Rule Symbols: f_1, g_1, h_1, p_1, q_1, +_2 Defined Pair Symbols: G_1, +'_2, F_1, H_1 Compound Symbols: c6_2, c16_1, c2_2, c3_2, c4_2, c7_2, c8_2, c9_1, c10_1, c11_1 ---------------------------------------- (63) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(G(z0)) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) S tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(G(z0)) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) K tuples:none Defined Rule Symbols: f_1, g_1, h_1, p_1, q_1, +_2 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1 ---------------------------------------- (65) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: F(s(s(z0))) -> c(G(z0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) S tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) K tuples:none Defined Rule Symbols: f_1, g_1, h_1, p_1, q_1, +_2 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1 ---------------------------------------- (67) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) S tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) Defined Rule Symbols: f_1, g_1, h_1, p_1, q_1, +_2 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1 ---------------------------------------- (69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(0) -> 0 f(s(0)) -> s(0) f(s(s(z0))) -> p(h(g(z0))) f(s(s(z0))) -> +(p(g(z0)), q(g(z0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) S tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_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. G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) We considered the (Usable) Rules:none And the Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = [1] POL(+'(x_1, x_2)) = 0 POL(0) = [1] POL(F(x_1)) = [1] + x_1 POL(G(x_1)) = x_1 POL(H(x_1)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c16(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(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(g(x_1)) = x_1 POL(h(x_1)) = [1] + x_1 POL(p(x_1)) = [1] POL(pair(x_1, x_2)) = [1] POL(q(x_1)) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1 ---------------------------------------- (73) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) by G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))), G(0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))), G(0)) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))), G(0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))), G(0)) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c8_2, c9_1, c10_1, c11_1, c_1, c7_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c8_2, c9_1, c10_1, c11_1, c_1, c7_2, c7_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) by G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))), G(0)) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))), G(0)) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))), G(0)) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))), G(0)) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c9_1, c10_1, c11_1, c_1, c7_2, c7_1, c8_2 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c9_1, c10_1, c11_1, c_1, c7_2, c7_1, c8_2, c8_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace H(z0) -> c10(+'(p(z0), q(z0))) by H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) H(z0) -> c11(+'(p(z0), q(z0))) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, H_1, F_1 Compound Symbols: c6_2, c16_1, c9_1, c11_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace H(z0) -> c11(+'(p(z0), q(z0))) by H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c6(H(g(z0)), G(z0)) +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, +'_2, F_1, H_1 Compound Symbols: c6_2, c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(z0)) -> c6(H(g(z0)), G(z0)) by G(s(0)) -> c6(H(pair(s(0), s(0))), G(0)) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(0)) -> c6(H(pair(s(0), s(0))), G(0)) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0))) -> c(H(g(z0))) by F(s(s(0))) -> c(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(0))) -> c(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(H(g(z0))) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c6(H(g(z0)), G(z0)) G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) G(s(z0)) -> c9(G(z0)) H(z0) -> c10(+'(p(z0), q(z0))) H(z0) -> c11(+'(p(z0), q(z0))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1 ---------------------------------------- (93) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: F(s(s(0))) -> c(H(pair(s(0), s(0)))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) by F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) by G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) by G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) by G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) by G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) by G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) by G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) by G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) by G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) by G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) by G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) by G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) by G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c10_1, c11_1, c6_2, c6_1, c_1, c7_1, c8_1 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) by H(pair(z0, z1)) -> c10(+'(z0, z1)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c10_1, c11_1, c6_2, c6_1, c_1, c7_1, c8_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) by H(pair(z0, z1)) -> c10(+'(z0, z1)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c11_1, c6_2, c6_1, c_1, c7_1, c8_1, c10_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) by H(pair(z0, z1)) -> c11(+'(z0, z1)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, H_1, F_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c11_1, c6_2, c6_1, c_1, c7_1, c8_1, c10_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) by H(pair(z0, z1)) -> c11(+'(z0, z1)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) by G(s(s(x0))) -> c6(H(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), G(s(x0))) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(h(g(z0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(s(z0)))) -> c(H(h(g(z0)))) by F(s(s(s(x0)))) -> c(H(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) by F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) by F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) by F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) by F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) by F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) by F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) S tuples: +'(z0, s(z1)) -> c16(+'(z0, z1)) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: +'_2, G_1, F_1, H_1 Compound Symbols: c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(z1)) -> c16(+'(z0, z1)) by +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1, c16_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: G(s(0)) -> c7(+'(p(g(0)), s(0))) G(s(0)) -> c8(+'(p(g(0)), s(0))) F(s(s(0))) -> c(+'(p(g(0)), s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c9_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) by G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))), G(s(0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(z0)) -> c9(G(z0)) G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))), G(s(0))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c9_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace G(s(z0)) -> c9(G(z0)) by G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))), G(s(0))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1 ---------------------------------------- (153) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) by G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) by G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c7(+'(s(0), q(g(0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(0)) -> c7(+'(s(0), q(g(0)))) by G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1, c7_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace H(pair(z0, z1)) -> c10(+'(z0, z1)) by H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c10(+'(z0, z1)) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1, c7_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace H(pair(z0, z1)) -> c10(+'(z0, z1)) by H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c11_1, c16_1, c9_1, c1_1, c7_1, c10_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace H(pair(z0, z1)) -> c11(+'(z0, z1)) by H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) H(pair(z0, z1)) -> c11(+'(z0, z1)) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1)))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, H_1, +'_2 Compound Symbols: c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c11_1, c16_1, c9_1, c1_1, c7_1, c10_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace H(pair(z0, z1)) -> c11(+'(z0, z1)) by H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) G(s(0)) -> c6(H(pair(s(0), s(0)))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1)))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, +'_2, H_1 Compound Symbols: c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c16_1, c9_1, c1_1, c7_1, c10_1, c11_1 ---------------------------------------- (173) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(s(0)) -> c6(H(pair(s(0), s(0)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: g(0) -> pair(s(0), s(0)) g(s(z0)) -> h(g(z0)) g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0))) h(z0) -> pair(+(p(z0), q(z0)), p(z0)) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) p(pair(z0, z1)) -> z0 q(pair(z0, z1)) -> z1 Tuples: G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0))) F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0)))) G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0)))) G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) G(s(0)) -> c8(+'(s(0), q(g(0)))) G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0)))) G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0)))) G(s(s(x0))) -> c6(G(s(x0))) F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0))))) F(s(s(s(s(z0))))) -> c(H(h(h(g(z0))))) F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0))))) F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0)))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))))) F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) F(s(s(0))) -> c(+'(s(0), q(g(0)))) F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0)))))) +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))))) G(s(s(0))) -> c1(G(s(0))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))))) G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0)))))) G(s(s(0))) -> c1(H(h(pair(s(0), s(0))))) G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0))))))) G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0))))) H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1)))) H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1)))) S tuples: +'(z0, s(s(y1))) -> c16(+'(z0, s(y1))) K tuples: G(s(s(y0))) -> c9(G(s(y0))) G(s(s(s(y0)))) -> c9(G(s(s(y0)))) G(s(s(0))) -> c9(G(s(0))) G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0))))) G(s(s(s(0)))) -> c9(G(s(s(0)))) Defined Rule Symbols: g_1, h_1, +_2, p_1, q_1 Defined Pair Symbols: G_1, F_1, +'_2, H_1 Compound Symbols: c6_2, c_1, c7_2, c8_2, c8_1, c6_1, c16_1, c9_1, c1_1, c7_1, c10_1, c11_1